Is it possible to upgrade Mac Pro 2.1 to 5.1 firmware? Does it also make sense -- is the hardware different? Has it been done before?
No, it is not possible, so it doesn't make sense and hasn't been done before.
Yes, the hardware is different (considerably). It's a completely different board and layout where the CPU and RAM are. The difference on the cases is around the FireWire 400 ports.
The 1,1 could be flashed to 2,1, and the 4,1 to 5,1; that mainly buys you newer CPU support (well, newer compared to the original firmware's capabilities).
The 1,1 and 2,1 have 32-bit EFI firmware; many have searched for ways to change that without much success (see MacRumors' forums). One answer I read suggested swapping out the motherboard.
All three (1,1-3,1) use older DDR2 frame-buffered DIMM's, whereas the 4,1/5,1 use DDR3 DIMMs. The supported CPU's are also different.