tigersty1e, the 360 is optimized for 16:9 TV resolutions. It'll output it's closest resolution and your monitor will scale it, if it can. It'll look a little fuzzy if you use a stretch mode, but it'll be pixel-perfect if you do a "centered timings"* mode (the xbox resolution best for your monitor will be 1280x720 scaled up. The newest update may even actually support 16:10 resolutions but I could be (and usually am) wrong.
Gaard, running through a VGA splitter/switchbox is going to degrade the signal. I don't recommend it, but it can be done.
As for your second point, you'd need a set of speakers with a set of red/white RCA inputs - or an RCA-Stereo minijack converter for plain old computer speakers.
Thirdly, the VGA switchbox will have two(or more) inputs that you connect the 360/computer to, and one output that you'd connect the monitor to. The box takes care of the gender changing for you.
*I don't know what this mode could be called on your monitor. It would be the mode that keeps the black bars all the way around, only using the actual pixels requested by the input signal.