Core 2 Duo is a 64 bit processor, yes.
You do not need a 64 bit version of Windows Vista... 32 bit will run fine on Core 2 Duo. And for now I highly suggest you get the 32 bit version.
When you buy Vista, have the option of 32 bit or 64 bit, but not both. HOWEVER, if you buy the full RETAIL 32bit version of Vista, you can send in $10 in to Microsoft and get a 64 bit copy. I recommend going this route.
64 bit Vista drivers are very scarce right now. I had to ni*gger rig 64 bit Vista to work and it took many, many hours. It's not worth the hassle. Eventually drivers will be plentiful and 64 bit will be fine... but not for another few months.