They don't affect the carry flag because the zero flag does pretty much the same thing, allowing data in the carry flag to be preserved. For inc reg₈, the zero flag is affected in the exact way the carry flag would be affected, both of which would be set if the register was 0xFF before the operation. For dec reg₈ it's not exactly the same, but it's very close. The zero flag would be set if the register was 0x01 whereas the carry flag would be set if the register was 0x00, but this can be easily corrected for by keeping the register's value at 1 above the value you actually want it to represent.

Yes. There's no reason to affect the carry flag if you can look to another flag for (nearly) identical results. I've seen a few routines that rely on this lack of carry flag affecting to function optimally.

The inc reg₁₆ and dec reg₁₆ instructions don't affect flags because they are very often used to simply advance a pointer in data as part of a routine that needs the flags for other information. Also, it's simple enough to make your own version that affects flags by using 8-bit inc or dec instead.