Another 32 bit/64 bit issue

All Linux OS come with two types of ISO, i386 and amd64 or x86_64.

Why is the word AMD appended before 64? Aren't Intel's Core2 family also 64 bit? My computer is a Core2Quad, isn't that 64 bit? At the moment, I am running Windows on it and it is probably 32 bit version. The 64 bit software don't work on it for that reason?

I've heard 64 bit is for 4GB+ systems. Then is 32 bit just fine for my system, no matter which OS I use?

What is good for Core2Duo and Core2Quad? 32 or 64?
 
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