Re: 64 bit processor, 32 bit cpu
Most of the time, no.
The vast majority of programs are still 32-bit, so running a 64-bit OS won't speed them up unless it takes up more RAM than a 32-bit OS can address (when it would have to start using paging file)
But if the program is compiled in 64-bit, it will run faster than if it was compiled in 32-bit; 64-bit code runs faster, though takes more memory.
If you had a 64-bit program, you would have to be running a 64-bit OS to use it; it wouldn't work in a 32-bit OS.