If a good is stolen, does the previous possessor have an indefinite immediate right to possession of that good? (Until it's returned)
In all jurisdictions, civil claims (including torts) are subject to statutes of limitations. For example, in new-south-wales, this is covered by s21 of the Limitation Act 1969. The specific tort here is detention of goods [detinue] (conversion is only applicable against the original thief) and the limit is "six years running from the date when the first cause of action first accrues to the plaintiff or to a person through whom the plaintiff claims."
The cause of action would accrue when A became aware or should have become aware that M was in possession. On the facts, A has six years from the date of being notified by the transport department. However, if A had (or should have had) knowledge that J possessed it (or any previous possessor did) then the limitation period would run from that time.
Essentially, the purpose of such statutes is to force people to use or lose their rights.
1. For all that time between 2005 and 2015, did A have a superior right to possession over those who seemingly legally bought the motorbike, but only had actual possession?
Subject to the above, and in most cases, yes. See If a stolen item is stolen again, who has the liability?
2. Is it likely that a mandatory injunction for the return of the bike would be awarded here?
Injunctions are only used when damages would be an insufficient remedy. For this to be the case, the bike would have to have some quality that made it irreplaceable. For example, if it was unique in some way - the bike that won the 1922 grand-prix or the bike that A's dad gave to A on A's graduation.
On the facts as stated, it appears that A could be "made whole" by monetary damages.
The value of the damages payable to A is the value of the goods at the time of conversion (IBL Ltd v Coussens  2 All ER 133). Specifically, this is the value of A's loss which may not be simply the market value of the goods.
For example, if the market value of the bike at the time was $2,000 but A could prove that he had had an offer to buy the bike for $4,000 thin A's loss is the value of the lost contract. Similarly, A would be entitled to the costs from 2005 to today of hiring a substitute chattel to replace the stolen one if that was a loss that A had incurred.
It's likely that a court would refer the parties to mediation to see if they can agree on a settlement. Only if that fails would the court impose a judgement.
However, for simplicity, let's assume that the figure of $3,000 is the ascertained damages then M would be ordered to pay A $3,000 and A would be ordered to transfer the bike to M.
3. Is it true that there would be no damages awarded given the increase in value of the bike?
As stated above, this would only arise if A retained ownership which is not a likely outcome. For the sake of discussion, let's assume that the bike was unique in a way the damages would be an inadequate remedy and the court awarded ownership to A.
M would potentially have a claim on A for unjust enrighment on the grounds of ignorance. However, in the case of a motor vehicle where there is a government register which can be relatively easily checked to verify title, this claim is likely to fail. For other chattels (e.g. furniture) where verifying title is next to impossible, M might have better prospects.
4. Would J be compensated by the court for the money spent on the bike?
5. Have M and J converted the motorbike (against A) by selling/buying the bike?
M & J did not take the bike from the rightful owner so they have not converted it. If M hold out against A's rightful claim of ownership then M has detained it.
M can seek restitution from J for their loss because J sold the bike without having good title to it. Depending on the jurisdiction, this is either a breach on an implied term of the contract, a breach of a statutory guarantee that sits beside the contract, or both.
Similarly, J has a claim on their vendor and so on up the line until we come to either the actual thief or a person who knew (or should have known) that they were buying stolen property. That person has no claim on anybody and is also subject to criminal sanction, however, it is likely the statute of limitation has run on that.