If I made an Xbox 360 game with XNA, would I need to debug it on the Xbox 360 as I was coding it? Or would there be a small emulator on my computer to debug it?
To debug on your PC you can set up an additional windows game project (XNA runs on Windows as well). Then you would have two projects, one XBox and one Windows that run the same game. You just want to make sure that your solution is set up to be cross platform. The only problem with this is that it doesn't emulate an XBox entirely as in you aren't limited by hardware in any way and as such you have a good chance of experiencing bugs on your XBox that you can't reproduce on your PC.
Yes, you don't need an XBox 360 to run or debug an XNA game. In fact, even if you have a developer account and are running it on an XBox360, you would still debug on a PC.
We have the professional dev kits in our studio which give you extra power to compensate for the extra bits you run while in debug builds, but we STILL debug on PC.
Also just to make it super clear - you can use a normal 360 to test and run games you make, it doesn't have to be a dev kit.
To debug an XBOX 360 game you need a devkit and here is how you can buy one.
You can run the game on your standard XBOX 360 if you have payed a subscription for your developer account.