Basically CPUs, GPUs and FPGAs all converge to the crab equivalent of computation. They all expose the same capability with different areas of optimization.

All logic can be implemented with memory, and all memory can be implemented with logic (in some sort of feedback configuration).

Neither is typically done in practice except for specialized purposes like FPGAs and the instruction described in this article. High-speed registers and static RAM are sometimes built with logic but it's more common to build them directly with transistors than with gates.

Most but not all. Actel/Microsemi use a small tree of muxes and gates.

So does the 74181 ALU.