Under libertarian homesteading, how is the physical height and depth of a claim on the Earth's surface determined?
For example, does a homesteaded plot only go as deep underground as work is done (i.e. planting, foundation construction, etc...) or does a homesteader own the mineral rights under his plot in a cone shape down to the Earth's core? If it's the former, are other landowners free to tunnell under the plot, and are they fully liable if they cause a cave-in which damages the surface homesteader's land?
Conversely, how are air rights determined? Is it a violation of the surface homesteader's property rights if someone builds an overhang on their adjacent property or parks a blimp overhead blocking out sunlight? Would flying an airplane through the air constitute homesteading, giving the airlines or the first civil pilots ownership of airspace along their flight paths?
If you read your post as satire, I think it answers your own question. In other words, those things are for the market to decide.
Why anarchy fails
What AJ said. There aren't any magical answers to your questions; market participants would decide everything.
Well, of course it’s for the market top decide but I myself would go decisively with “you homestead only what you use” i.e. no air, no underground, no mineral rights. The bare minimum. Air routes are actually homesteaded by airliners.
If the market is to decide how one must homestead in a objectively acceptable way, then that means that the meaning of the non-aggresson principle, (which is bases on property rights, which in turn is based on the homesteading principle and contracts of exchange), must be determined by the market. So the principle defining the free market must be determined by the market.
Better word choice than "market" is "natural order." Natural order isn't defined by the NAP, although if we are to say that most people believe the NAP is the highest principle, the natural order will tend to uphold it.
See: http://mises.org/journals/scholar/hasnas.pdf
Indeed. Nothing, not even the free market, is god-given. In the long run we adjust to what works. It just so happens that in this universe free markets work, so we stick to them.