First question. When you buy some land. How far can you dig before you are not on your land anymore? Or am I a land owner in Russia? Assuming you can dig without problems.
Second question. Let’s say I drill for oil on my land and find some, however the same supply of oil is under someone elses property too. What happenes then? Is it all mine if I can drain it first?
Subsurface property rights extend down only to the center of the earth.
Mineral rights are different. I can only speak for Oklahoma law, since I used to work for my dad doing title opinions for oil companies researching exactly who had to be compensated for oil production in a given area. This is dictated by “pooling” and varies from one area to another, as determined by the Oklahoma Corporation Commission (which regulates everything energy/utility-related here). Often, it goes by sections (a square mile), but there are a vast multitude of variations. In addition, not all property owners have the mineral rights to their property. These can be sold seperately or retained by the original landowner. IIRC, Texas and Kansas law (we did a few in those states, too) are very similar.
Sewalk has answered your question. Vision your property as a triangle, with the apex at the center of the earth. However, it does not extend infinitely into space, only so far as you can make reasonable use of it, whatever that means. In any event, airplanes fly over it without trespassing, and don’t lay claim to any portion of the moon.
Mineral rights can be severed from the fee simple and, in mineral areas, this is often done. You have to read the legal description of your property to see if the mineral rights have been excepted. If they have, there is a separate chain of title covering them.