Is the government allowed to call its property "private property", even though we the taxpayers paid for it?

Yes they can and they do.

They can restrict its use. And, in many cases, they have to restrict its use. For example, they can't let you wander around somewhere that they are storing bombs or radioactive materials, for obvious reasons.

All government property is public property. However, because of "national security" or "public trust" or whatever, the government can restrict lands it is the caretaker of.

One of the biggest contradictions among "public lands" are Wilderness Areas which are regulated to keep the public out.

I am a victim of eminent domain. To build a dam and no dam was ever built. Land speculation, pure and simple.

They can restrict access and use.

We should be able to walk into Area51 and take a p*ss there if we like.