Can a 32 bit OS run in a 64 bit processor?
Is it possible to connect an HDD with a 32bit Windows 7 OS to run on a Z77 motherboard with corei7 3770K processor. Or will I need a 64bit OS?
No, you do not need a 64bit OS. A 32bit OS can run on 64bit hardware with no problem. You just won't get any advantages from your 64bit hardware.