You do not hook up the Xbox 360 with the PS3 they must be connected individually. They can be connected to a switching device and that connected to different TVs if it is complex enough of a switching device to allow the switching at both input and output.
I don't even know how you could play one on Xbox 360 without a device similar to the Kinect. But to answer your question, not at the moment.
Output. You insert a CD and show (play) the movie. A DVD Multi Recorder is an output and input device. You can insert a blank CD into it and record even pictures. Then you can click on play and play out of the CD right after you record into it.
No PS2 games do not play on the Xbox or Xbox 360
No
you need a upgrade
Unfortunately, no it can not.
Evidently you need a device supporting VGA capture and recording with high resolution from the XBOX 360 output. An Epiphan's capture card VGA2USB LR with lossless signal compression seems to be a convenient solution for your needs
A long time ago, you could only play some xbox original games. Now, I believe you can play any original xbox games.
Yes you can. All versions of the xbox 360 are compatible.
you could play any game
In order to play on Xbox Live one only has to connect their Xbox to a wireless network. An Xbox can be connected to a network through the settings on the device. If there is no router available an Xbox can be connected to a Windows computer and use that computers internet access.