Real, official R4s can run a few firmware.
I -know- that they can run the
official firmware, the
R4li firmware and
YSMenu as well as
Moonshell2.
R4li has better ROM compatibility than the official firmware, but has a pretty major sound bug where volume plays at a low level and is NOT effected by the volume control (at least on a DS "phat"). YSMenu typically has better compatibility than R4li, but R4li does run some games that YSMenu won't.
Supposedly, the official R4s can run any firmware from cards that have been cloned from the R4 if you rename and encrypt the .DAT file, but I have not tested it personally so I cannot give a better answer. This seems to be contradictory as the R4 cannot run the DSTT firmware, which I believe is an R4 clone.
However it should be noted that not all "R4"s are clones of the original R4. Some of them are AceKards.
As for the opposite, I don't believe the official firmware will work on clones.
YSmenu will run, but not as firmware, though you can set it up to auto-boot on startup.