Ruling is done by rulers, and the people who have lots of power are the powerful people. Money can be exchanged for goods and services.

If we have no way to determine whether two accounts are held by the same person, so that influence can only be determined by what one has, then it should behave the same pattern I would think.

You can use weapons. Let's call it Proof of Ammo.

Depends on what properties you want your system to have. For some properties yes, for others no.

Of course anyone can come up with something new at any time.

Yes, see the Bitcoin white paper ;)

Proof of stake isn't a government, it's a network time protocol.

Proof of Eyeball —sama

Yes. Proof of Logits.

but that just how life goes