1. Yes, you can record the audio output by using the headphones jack, or you can use the USB audio if it works for you-- some people have had trouble with the driver.
2/3/4. You can use the PSR-E363 as a MIDI keyboard controller by connecting it to your computer by a standard USB 2.0 cable, often called a "printer cable." The PSR-E363 is class compliant, so you may be able to use its USB MIDI connection without installing a driver, but if you do need to install a driver it should be the Steinberg USB driver that's on Yamaha's web site. You'll need a VI (virtual instrument) or soft synth (software synthesizer) that has the sound you want to record. Some VIs and soft synths are standalone programs that include their own "host" program, while others are designed to be used as "plug-ins" inside of a DAW (digital audio workstation software) or other VI host program. If you're going to be recording the VI then you'll probably be using a DAW. The standard procedure is to add a new MIDI track in the DAW, then assign the PSR-E363 or other controller as the MIDI Input device, and assign the VI you want to use as the MIDI Output device. Some DAWs call this an "instrument track" rather than a MIDI track.