I know that you mean a system that transfers money but you are also describing Anubis because PoW is literally to make accessing the site cost more and scale that cost proportional to the load.
Sort of. Anubis is frontloading the cost all at once and then amortizing it over a large number of subsequent requests. That detail is what's causing the issue when browsing with additional privacy measures.
No, cost is used in the fullest abstract meaning of the word here.
Time cost, effort cost, monetary cost, work cost, so long as there is a functional limitation that prevents resource exhaustion that is the point.