Landowners generally control the air above their land to some certain limit, which is why you can fly your drone at 2000' elevation over their land but not necessarily 10' elevation over their land.
The situation is a bit confusing but the general idea is that altitude doesn't matter, regulatory context does. The FAA holds the exclusive right to regulate civil aviation and so property owners are not able to restrict any aviation activity unless the FAA permits it. There are aviation regulations that generally prohibit operation of aircraft within 500' or 1000' feet of the ground (depending on unpopulated or populated area) except when on the way from or to it, but there are plenty of nuances to that rule (most notably that it's all completely different for sUAS) and it's nothing to do with property ownership but rather aviation and public safety.
In other words, property owners are generally viewed as owning the airspace all the way up from their land, but that ownership does not confer any rights to restrict aviation, which is all done according to a separate body of federal regulations that has very little interest in land ownership. This is a similar situation to mineral rights, where land ownership matters more but there is still a separate system of regulations and deeds that operates largely independently from property title (which is why it's common in rural areas to buy land where the mineral rights are held by someone else).
Notably, though, this is all about flying - ground operations are different and you must have permission of the property owner for ground operations (except in emergencies when the regulations are generally tossed out the window for better glide). So property owners can restrict takeoff and landing, and (I think this is a little bit fuzzier legally but still generally agreed) remote operation of aircraft from their land.
Incorrect. You can't fly your drone at 2000' AGL in almost any circumstances, anywhere, ever. Elevation is also the wrong term. Drones are capped in the US at 400 feet above ground level, with exceptions for part 107 pilots within 400 feet of a structure or terrain feature, and specific situational waivers.
There is no specific rule against flying even 1 foot above ground over someone else's property, though it's very unlikely that doing so would not violate other rules unrelated to airspace. These include operating a drone unsafely, violating laws against harassment, and photography in a context where someone would have a reasonable expectation of privacy.
