How to capture (high definition) audio and video from the XBox 360

The Roxio Video Capture USB and Dazzle Video Capture devices allow you to redirect the RGB output from video game consoles to a computer.

This product from Blackmagic Design seems to record RGB, audio output, S-Video, and HDMI.

From there, you can record your PC screen, either with drivers from the companies of the products or with alternative screen-recording software.

(Note: These solutions cost money to buy the products, but work for any devices that output via the formats listed.)

Your first question was asked and answered here,
your second one here,
your third is not really appropriate, it's too localized and also subjective, there's not a real answer I can give you,
and your fourth is answered here.

Please read those, you can comment on them if you still have any questions about those topics.