Can I pay some money at my local apple store to get my iPhone 4 replaced with a new one?

You won't be able to get a new one, probably refurbished, but it would be an OOW replacement if something is wrong with the phone. I believe the OOW price for the iPhone 4 is $149USC. What is wrong with the phone that you want to replace it, maybe someone here can help you get it working again.

It's possible:

iPhone Service Support and Costs

Yes. But the program is not a trade-in program. It's an exchange program for non-functioning or damaged devices. If your is working they may choose not to exchange it. You'll have to ask what they will do for you.

Yes, they should replace it. But that's not my call.

If you have performed all of the user troubleshooting http://support.apple.com/kb/TS2802 and it does not function, the would probably replace it as an OOW as described above.

No, only like version, color and capacity.

Only way to get a white one is to sell your black one and buy a white one.