The 6502 gets by doing immediate load: 2 clock cycles, 2 bytes (frequently followed by single byte register transfer instruction). Out of curiosity I did a quick scan of the MOS 1.20 rom of the BBC micro:
LDY #0 (a0 00): 38 hits
LDX #0 (a2 00): 28 hits
LDA #0 (a9 00): 48 hits(You can squeeze in a cheeky Txx instruction afterwards to get a 2-or-more-for-1, if that would be what you need - but this only saves bytes. Every instruction on the 6502 takes 2+ cycles! You could have done repeated immediate loads. The cycle count would be the same and the code would be more general.)
Hard to tell, but I don't think so ;-)
I suppose using Txx instructions rather than LDx is more of an idiom than intended to conserve space. Also, could an LDx #0 potentially be 3 cycles in the edge case where the PC crosses a page boundary? (I'm probably confused? Red herring?)
The full 16 bits would be incremented after each instruction byte fetched, and it didn't cost any extra if there was a carry out of the MSB.
MIPS - $zero
RISC-V - x0
SPARC - %g0
ARM64 - XZR
Even tiny tiny CPUs can do sub in one cycle, so I doubt that. On super-scalar CPUs xor and sub are normally issued to the same execution units so it wouldn't make a difference there either.