Who Is On The Risk?
This depends upon the language of the respective Gecko and NoState policies, the liability of each insurer is determined independently, and it is possible that neither, one or both of them would be "on the risk" as they say in the industry.
Usually, insurance policies are drafted on either an "occurrence" basis (i.e. when the "incident" occurred), or on a "claims made" basis (i.e. when the "incident" was reported to the insurer). Unless the homeowner was awfully mindful and called his insurance agent almost simultaneously with the fire department, NoState would be on the risk if it was a claims made policy, and Gecko would not be on the risk if it was a claims made policy.
In ordinary U.S. homeowner's insurance policies, an occurrence policy is much more common than a claims made policy. So, one would first look at the definition of an occurrence in the Gecko policy and see if there was an occurrence prior to the expiration, in which case the Gecko policy would cover it, and then one would look at the definition of an occurrence in the NoState policy and see if there was an occurrence after the policy term commenced, in which case NoState would cover it. The same principles discussed in this answer would apply in a non-U.S. case, but the typical language found in a homeowner's policy in non-U.S. jurisdictions might very well be different.
There is a very decent possibility in this situation that this would count as an occurrence for both Gecko and NoStates' policy, although if only one of the policies was on the risk, it would probably be the Gecko policy and not the NoState policy.
There is no one uniform definition of occurrence, many definitions of when an occurrence takes place are not sufficiently precise to exclude either Gecko or NoState in this fact pattern, and generally speaking, there is a legal rule that interprets legal language in an insurance contract in favor of the insured to the extent that there is any doubt as to its meaning.
If both policies were on the risk, they would each have to provide coverage to the extent of their contract, although there would usually be some anti-double recovery language in each policy to prevent the insured from getting more payments for any particular kind of loss than the total losses of the insured. The relative burdens of each insurer would be worked out in a negotiation involving the Gecko, NoState and the insured, and by a court if the negotiations did not result in an agreement. A negotiated deal would reflect any uncertainty regarding the likelihood that Gecko or NoState respectively would be required to pay.
For example, if Gecko was clearly on the risk according to its policy language, and NoState was only probably on the risk but not clearly based upon its policy language, in double recovery situations, NoState might pay less than 50% of the portion of the loss that would otherwise give rise to a double recovery.
What Is Covered?
The amount of coverage (e.g. replacement value v. current value of the building and property therein, temporary accommodations, etc.) would depend entirely on the coverage set forth the insurance contract of each insurance company that is on the risk. Coverages vary a great deal and influence insurance premium prices. Usually, for example, you would be able to get policies with or without temporary accommodations coverage as you wished when you purchased the policy and that decision would govern what the insurance company on the risk would have to pay for when a claim was made by the insured.
As an aside, keep in mind that a significant part of the payment would go to the mortgage holder, if any, and that a homeowner's insurance policy almost never covers the part of the value of a home that is attributable to the land that it is build upon, rather than that building that was destroyed itself.
If your house was worth $300,000 and subject to a $150,000 mortgage, and was built on a lot worth $100,000 if it were vacant, the insurance payout would typically be $200,000, of which $150,000 would be paid to the bank holding the mortgage and $50,000 would be paid to the home owner who would also retain title to the parcel of real estate that the house was built upon.
With the same facts, but a $250,000 mortgage, an insurance payout of $200,000 would go to the bank holding the mortgage, no payment would be made for damage to the house to the home owner directly, and the homeowner could continue to retain the title to the parcel of real estate that the house was built upon subject to a remaining $50,000 mortgage balance. What what would happen next would depend upon the fine print in the mortgage.
The amounts paid to the mortgage holder, as well as the amounts paid to the home owner, would both contribute to the amount of the insurance company's subrogation rights discussed below.
Bonus question - can either/both Gecko or NoState sue Samsing?
Any insurance company who pays an insurance claim caused by the tortious fault of another (e.g. Samsing, if its product was defective in a manner sufficient to give rise to legal liability) has a right to sue the party at fault for the loss the insurance company suffered in what is called a subrogation lawsuit.
If both companies paid, both would have subrogation claims against Samsing for the amount that they paid.
The insured could also sue Samsing for any losses suffered, whether or not they were covered by insurance, but any insurance company that paid a related claim would also have a lien on any recovery of the insured in a suit by the insured against Samsing, for any loss paid by the party at fault to the insured that was within the scope of what the insurance company paid the insured for.
If the insured had losses beyond what was covered by the insurance payment, or losses different in kind from what was covered by the insurance payment, and the insured recovered damages from the party at fault for those losses, this additional recovery would not be subject to the subrogation lien.
For example, damages recovered from the party at fault for emotional distress, pain and suffering, permanent physical impairment, and temporary accommodations (if none of the insurance policies on the risk covered temporary accommodations) would not be subject to a subrogration lien of the insurance company or companies that paid claims to the insured for other damages suffered by the insured in connection with the incident.
Also, if, for example, the insurance policy had a policy limit of $100,000, but the insured suffered $150,000 of damages and recovered that from Samsing in either a collected court judgment or a settlement payment, the $50,000 of excess recovery for economic damages beyond the subrogation lien would belong to the insured.
Since the insured would typically sue with a lawyer hired on a contingent basis, usually, the insured lawyer would reach an agreement with the insurance companies holding the subrogration liens for them to pay a contingent fee to him on the same basis as his primary client if he recovered funds subject to the subrogration liens, so that his client would not walk away empty handed after using his entirely recovery to pay his attorneys' contingent fee, or would reach some other agreement addressing the same problem.