Would a 64Bit processor work with a 32Bit operating system?

It can do but it is really the other way around. A 32-bit operating system can work with a 64-bit processor. However, if the processor doesn't have a "soft" switching mode, you will have to manually switch the 64-bit processor to 32-bit mode via the CMOS setup.

Ideally you should install a 64-bit operating system and leave the processor in 64-bit mode. This will then allow you to run 32-bit programs and 64-bit programs side-by-side.