The majority of the clones are using "NES On a Chip" or "SNES On a Chip" - they have the same issues as the original minimal requirements based emulators and are generally quite quirky.

Though it looks like at least one person is attempting to create a better one using FPGA: http://cegt201.bradley.edu/projgrad/proj2006/fpganes/

Yes, or replicate all of the original SNES hardware in one FPGA chip (and perhaps a few external passive components).

Citation needed, but pretty sure I heard of compatibility issus that the clones had.

As mentioned in other comments, some SNES cartridges themselves contain coprocessors and other hardware. A clone console doesn't need to do deal with this since you're still playing physical cartridges (versus pure software ROMs).

But clone systems do have issues with many games anyway...