SKUs arn't generally locked to hardware types so switching from 7 pro to 7 home should be doable.
Switching from 32 to 64 bit ... depends on the system. Most systems in the last 7 years or so should be x64 compatible - you can check this with a tool like cpu-z which gives processor information - it should have a processor flag like AMD64 or EMT64 to be able to run a 64 bit OS. If it does, it should be able to run a 64 bit OS with no issue
This would be an example of a system that came with a 32 bit OS, where i could install a 64 bit OS - in this case XP to windows 7. Assuming your other hardware is supported, it should not be a problem to switch from 64 bit to 32 bit, or vice versa.
While you'd want to check if your OEM has 64 bit drivers, chances are that what you want would work quite easily.