Well here's my situation.
I bought an 1st Gen 4GB in the market place. The phone I bought was new because they guy who sold it to me took it in and they gave him a new one since
it was still under warranty. I just noticed right now that the black part on the bottom back of the phone is a little loose and makes a clicking sound like it's not
in securely. I assume it's still under warranty. But since I'm not the original buyer of the phone would I still be able to get it fixed since it's still under warranty?
I have the Genius Bar paper that the original owner received when he took in the broken phone for a new one.

I only have the Genius Bar paper. Do they give you a new iPhone for this kinda stuff or does the
phone have to be severely damaged to get a new phone?
And do I have to make an appointment?

Go in there, and tell them that you bought the phone and it was replaced once but you don't remember when, and show them the paper. They should either replace it or they will do maintenance on the phone to repair or fix it. As long as it's not damaged or user damage then it should be under warranty.