This is all purely academic. There aren’t any ‘countries’ on Earth which aren’t inhabited, and there haven’t been since long before the first bandit decided to move in with his victims and establish a state.
Sure there are. A large proportion of the American West is essentially vacant land. That’s partially because much of the land is more or less useless, but there are many potentially useful parts which the federal government simply refuses to allow anyone to use.
As for the OP:
The only function of property rights is to determine who should be allowed to use scarce resources. If there’s some parcel of land, and the lumber jacks want it for lumber, and the hunters want it for hunting, and both can use it at the same time without interfering with the others, then there’s no need for property rights. If both cannot use the land without interfering with the others, then and only then is there a need for property rights. And if it turns out that way, the question is: whose claim is strongest? Who’s “using it the most?”
When comparing claims of use, one might consider temporal priority (who started using it first), or physical extent of use (especially with land), or investment in the property, or the value of the property to each party, or any number of other factors.
There is nothing in the NAP or any other aspect of libertarian ethics which can justify any one of these criteria for use over the others, as all of libertarian ethics presupposes property rights, and now we’re trying to justify original property rights.
We can be pragmatists here, at least until/unless someone makes a persuasive argument that universal adoption of one or the other criterion will yield the best results in general.
I would lean toward investment in the property: i.e. he who invests the most is the owner (measured in money). I tend to think this would be both fair (in some intuitive sense) and also likely yield the best economic results, as those investing more (one would assume) would be the best stewards of the property in the future. A good study of actual settlement of raw land would be useful in trying to see what the consequences of different homesteading theories would be. (and again, we are looking at consequences here [and maybe trying to draw a principle therefrom], not trying to find a justification for one or the other theory of homesteading within libertarian ethics, which is impossible, since it all preupposes property rights).