I use this very LCD monitor I'm using right now for my Xbox 360. It's great because i don't sit too far away from it when I'm playing as my monitor is only 15" (small bedroom too) so the size isn't an issue. Anyway, i use a Joytech Xbox 360 VGA cable to input the signal into my monitor (It's the same as plugging the VGA cable from the computer into the monitor).
15" monitors have a 1024x768 resolution, and in the settings of the 360 you can choose that as your resolution.
If you're interested in HD (like i am) and you cannot afford to buy a brand new HD TV, then it is definitely worth getting an Xbox 360 VGA cable whether it's the official one or the Joytech one, it doesn't really matter, you will still get a fantastic picture on all the games you play.
So, to answer all your questions:
1. It WILL work (if your monitor has a VGA input at the back) if not then you can get a gender changer for the VGA cable to sort that out.
2. You WILL see a hit in quality. The VGA cable produces the best picture available for the Xbox 360, no matter what resolution.
3. All Xbox games can play on a 1024x768 resolution, so you don't need to worry about some of your 360 games not working.
4. As i said, an Xbox 360 VGA Cable
5. I would highly suggest this.
Athlon.