Summary: Constitutionally, the power of provinces to control liquor sale and to apply sales tax to those same sales are distinct.
There must be sufficient presence in a province for its taxation power to apply. This is categorically not the case for sales aboard aircraft. For trains, case law is scant but a case-by-case analysis has so far been favoured. Independently, HST still applies according to the origin province, as that tax is backed by federal power.
Liquor control—including sale—is far more complex and is probably an open question. Federal jurisdiction over aviation and railways does not remove provincial powers with respect to control of liquor sale. However, those provincial powers might not apply in this context on their own merits, especially with regards to aircraft.
0. Answer overview
Due to the particularities of Canadian federalism, each combination of taxation & liquor control with respect to aircraft & trains could in theory have different answers. I've essentially reframed your specific questions into analysis of these four cases, though I provide direct answers at the end. I'm going to start with sales tax on aircraft since that has the strongest case law available. Then, I'll extend the reasoning to each of the other three cases.
Note that to avoid excessive caveating and further lengthening this answer, I've mostly avoided specifying how confident I am in my interpretations of the legislation and case law available. While I'm relatively well-read with respect to the Canadian constitution, I would not consider myself very knowledgeable in the areas of tax, commerce, transportation or liquor. Please make your own judgments based on the strength of the citations and arguments I present. In writing this answer, I generally got less and less confident the further along I was, the exact opposite of what happens after each drink aboard an aircraft (so I'm told).
1. Sales tax
Parliament is granted a very broad taxation power by the Constitution Act, 1867 s. 91(3), while provincial taxation power is limited to "within the Province" by s. 92(2).
In practice, sales tax in Canada is a
mess mix of systems. Parliament has established a federal sales tax, the Goods and Services Tax (GST). Concurrently, all provinces except Alberta have some form of provincial sales tax applied. None of the territories levy an additional sales tax. Further complicating matters, Parliament has developed a Harmonized Sales Tax (HST) whereby participating provinces have ceded their authority in favour of the federal government administering a single sales tax and remitting a percentage which the provinces specify. Currently participating are Ontario and the Atlantic provinces. In Quebec, the arrangement is mostly reversed with Quebec generally administering both federal and provincial sales taxes. The remaining provinces have separately administered federal and provincial sales taxes.
1.1 Sales tax aboard aircraft
There are two especially on-point Supreme Court cases, both involving specifically the sale of liquor in-flight. In The Queen (Man.) v. Air Canada,  2 SCR 303, the court found that:
Merely going through the air space over Manitoba does not give the aircraft a situs there to support a tax which constitutionally must be "within the Province". In the case of aircraft operations, there must be a substantial, at least more than a nominal, presence in the Province to provide a basis for imposing a tax in respect of the entry of aircraft into the Province.
In reaching this finding, similar American jurisprudence was cited specifically noting that "No question of legislative jurisdiction over air space was involved."
This reasoning was then extended to include stopovers. In Canadian Pacific Air Lines Ltd. v. British Columbia,  1 SCR 1133, this was further declared to apply to flights entirely within a single province. Therefore, provinces have no constitutional authority to tax sales in-flight as being in the air does not count as "within the Province" for taxation purposes.
However, the federal government is not so restricted. While GST is straightforward, things get interesting with HST. It is legislated for in the Excise Tax Act Part IX s. 165(2). Despite the remittance to the participating provinces, it is de jure a federal tax which can therefore still be constitutionally levied. The attached New Harmonized Value-added Tax System Regulations states in s. 23:
If a supply of property or a service (other than a passenger transportation service) is made to an individual on board a conveyance in the course of a business of supplying passenger transportation services and the property or service is delivered, performed or made available on board the conveyance during any leg of the journey that begins in any province and ends in any province, the supply is made in the province in which that leg of the journey begins.
Therefore, if the leg of a journey beings in an HST province, the sale of liquor in-flight is taxed according to the origin HST. Otherwise, only GST could be charged per constitutional reasons above. Note that Quebec falls in the non-HST category. While it administers both federal and provincial sales taxes, they are still separate taxes unlike the HST.
1.2 Sales tax aboard trains
The question is now whether sales aboard trains, and more generally, land-based transport are sufficiently "within the Province" to fall within provincial taxation power. Case law is scant, but four of the five applicable cases I could find have refused to mechanically apply the result from aircrafts, preferring to undertake a factual analysis on whether sufficient presence is established. They are Ace-Atlantic Container Express Inc, Re, 1992 CanLII 7103 (NL CA), Canpotex Ltd. v. British Columbia, 1999 CanLII 5844 (BC SC), Pétromont and Company Limited Partnership v. Ontario (Finance), 2008 CanLII 5980 (ON SC), Dow Chemical Canada Inc. v. R., 1992 CanLII 1019 (BC CA) and Lynden Transport Inc. v. British Columbia (Finance), 1981 CanLII 785 (BC CA).
Notably, Lynden involved only transit through the province and directly applied the result from R. (Man.) v. Air Canada to find that the province could not apply its taxation power in this instance. Canpotex also involved transit only, engaging in some analysis before considering itself bound by Lynden.
The other three are all similar fact patterns and results. Most operations of the taxed companies were transit through the province, but not exclusively. The non-transit operations were found to be a sufficient enough presence to allow the province to apply a general tax. Pétromont at para. 79 notably emphasized length of time railcars spent in the province as a reason for departing from R. (Man.) v. Air Canada.
Unfortunately, none of these are directly analogous to sale of liquor aboard a train. They were all about whether tax could be applied to the land transport vehicle or the goods contained therein. There are hints in some of these cases and their citations of CPA Ltd. v. BC that this may make a difference, but nothing anywhere near definitive.
Constitutional questions about provincial taxation power aside, I can find no statutory difference between planes and trains with respect to the applicability of HST per the Excise Tax Act regulations cited in the previous section (all cases cited here are pre-HST in the participating provinces).
2. Liquor control
There is no singular "liquor" power in the constitution. This has resulted in liquor control having an extraordinary amount of possible constitutional bases for both Parliament and the provincial legislatures. To maintain a degree of sobriety (I'm sorry), I will focus my analysis on the provincial power to control sale of liquor which typically applies within their borders and whether federal jurisdiction over aircraft and railways impede this power. To fully understand the scope of the constitutional complexity of liquor, I strongly recommend reading in full the lecture paper The Effect of Alcohol on the Canadian Constitution ... Seriously (2011) by the Honourable Morris J. Fish (Supreme Court justice 2003-2013) though it is not strictly necessary for this answer. Note there are a few paragraphs in French, they aren't crucial to overall understanding.
2.1 Liquor control aboard aircraft
As you point out, Air Canada v. Ontario (Liquor Control Board)  2 SCR 581 at least to some extent confirms that airlines can still be subject to provincial liquor control. Specifically, the court found at paras. 72-76 that provision of onboard liquor is not a core part of the federal aviation power. Therefore, this opens the door for provincial liquor sale law to be applied in-flight.
However, this provincial power must be grounded (again, I'm sorry) in a constitutionally granted power. In Air Canada v. Ontario the court acknowledged that a possible consequence of their main result was that provinces could impose importation fees on liquor that merely passed through their airspace (paras. 63-66). However, this is clearly dicta and was partially due to a very specific lending of federal power that isn't generally applicable here (or currently, as the involved federal statute has since been amended). I mainly wished to point out that the fact provincial airspace is not within provincial taxation jurisdiction does not imply provincial airspace is immune from all provincial authority.
Moving on to powers specifically granted to the provinces by the constitution, there are three different bases that justify the provinces' ability to locally control liquor sale enumerated in the Constitution Act, 1867 (emphasis mine):
- In each Province the Legislature may exclusively make Laws in relation to Matters coming within the Classes of Subjects next hereinafter enumerated; that is to say, [...]
- Shop, Saloon, Tavern, Auctioneer, and other Licences in order to the raising of a Revenue for Provincial, Local, or Municipal Purposes. [...]
- Property and Civil Rights in the Province. [...]
- Generally all Matters of a merely local or private Nature in the Province.
Those familiar with the constitution will recognize the latter two powers as the "default" ones which allow the provinces do most things within their territory. Indeed, Air Canada v. Ontario acknowledges that 92(16) authorizes the provincial liquor monopolies (para. 54). In addition to the two usual suspects, there's 92(9) which by its wording likely doesn't amount to full control, but certainly allows imposing licences on liquor sale. The main question is whether any of them can apply extra-provincially or in provincial airspace.
As you can probably guess by my added emphasis, extra-provincial legislation is generally not allowed (see eg. Reference re Upper Churchill Water Rights Reversion Act  1 SCR 297 at pg. 332). Yes, this does mean I'm extremely suspicious of WestJet's assertion tying their policy to a single province's laws, though I can certainly understand wanting to be in compliance with the liquor licences issued by your home province.
As for airspace, Air Canada v. Ontario appears to be the most recent Supreme Court decision addressing provincial airspace jurisdiction. All we can note is that the Supreme Court has acknowledged the possibility of provincial influence on their airspace. It's additionally possible that even if provinces do in fact have jurisdictional control, aircraft might not be found to be "in" the province much like they are not "within" for taxation purposes. Should the provinces not have liquor control jurisdiction on the aircraft, this jurisdiction would default to Parliament through the Constitution Act, 1867 s. 91 peace, order, and good government clause.
On the other hand, if the provinces have the necessary aircraft liquor control jurisdiction, it might be possible for Parliament to encroach on that jurisdiction, though I will decline detailed analysis. I would mainly be wildly speculating, and since your question concentrated on provincial power, I am content having found the necessary conditions for provinces to exercise their local liquor control power when there is no encroachment. I'll just briefly gloss the general constitutional principles: Parliament could not encroach by passing effectively provincial liquor control legislation and saying it's allowed because of their aviation power (pith and substance doctrine), though it's possible they could find a way through paramountcy or a separate power under s. 91 (in particular if sale aboard inter-provincial transport constitutes inter-provincial trade), since liquor control is complex and constitutionally divided.
Finally I'll quickly comment on Porter Airline's FAQ you cite. They partially attribute their in-flight drinking age to the Canadian Aviation Regulations. It's an incredibly lengthy regulation so it's very possible I missed something, but I can't find any basis for this attribution. Perhaps telling is that in the very next FAQ question on drinking limit, they state:
Under Canadian Aviation Regulations (602.04), cabin crew is expected to stop serving alcohol to a passenger who appears to be intoxicated.
A specific section of the regulations is cited, now that one definitiely exists! Also, this is a good example of a federal regulation I would guess to be constitutionally valid through pith and substance & paramountcy doctrines (again, assuming provincial liquor control jurisdiction to begin with). Avoiding having unruly intoxicated passengers can reasonably be tied to the federal aviation power and so would displace any provincial legislation contrary to that.
2.2 Liquor control aboard trains
While federal jurisdiction over aircraft is long established through the Aeronautics Reference, the basis for federal jurisdiction over trains is slightly more complex. The Constitution Act, 1867 s. 92(10)(a) confers federal jurisdiction over inter-provincial and international railways and s. 92(10)(c) allows Parliament to issue a declaration that a railway falls under federal jurisdiction, both in conjunction with s. 91(29). If Wikipedia is to be believed, railways are the majority reason for a 92(10)(c) declaration. The Queen v. Board of Transport Commissioners,  SCR 118 confirms that the federal jurisdiction over railways extends to any train on those railways.
I don't think it's controversial to state that provincial liquor sale law applies on trains outside federal jurisdiction. For those in federal jurisdiction, I don't see a reason why the legal reasoning in Air Canada v. Ontario wouldn't also apply, as well as the constitutional principles I covered in reference to it (substituting "airspace" for "railspace," I suppose).
For trains, I would just add that as land-based transport, they might be more likely found to be "in the Province" and therefore under provincial liquor control. Indeed, it's difficult to imagine that if they can be "within the Province" for taxation purposes, they wouldn't also be "in the Province," as that appears to be a less stringent requirement per the line of case law established by the cited R. (Man.) v. Air Canada, CPA Ltd. v. BC and Air Canada v. Ontario (though frustratingly, never an absolute direct constitutional confirmation).
3. Answers to your main questions
- Do provincial laws on sales and supply of liquor apply to services of alcohol on board a plane or a train subject to federal jurisdiction? If so, which province's law apply, and to what extent?
Open question, more likely yes on trains. If the province's law applies at all, it would apply within its borders and probably close to full extent.
- Does the law treat aircrafts (in air) and trains (on ground) differently? Does it matter if the plane or train makes any stop in the province?
Aircraft and trains can be treated differently for sales tax and maybe for liquor control as well. Essentially this comes down to land-based transport being "closer" to usual provincial jurisdiction. For aircraft, stops matter only for determining whether HST applies. For trains, stops could additionally contribute to them being found to be more fully in the jurisdiction of the province they stop in.
- Which province's sales tax (HST/PST/QST), if any, apply to in-flight and on-train purchases? I would assume the destination since that's the assumed place of consumption, but it seems that it would be very complicated for trains.
For travel by either mode, only origin HST (or federal GST) can be applied for each leg of the journey. PST and the provincial portion of QST are constitutionally barred from applying on flights. Provinces might be able to charge on-train sales tax within their borders given sufficient company presence.