1) no you don't need one. All digital pianos will have an MIDI In/Out (I/O) whic you can use to connect to the computer
2a) You'll need a sequencer to take in MIDI (ie cakewalk, cubase, pro logic). Note your sound card must also support MIDI I/O. simply just connect from your Keyboard MIDI Out to your soundcards MIDI In. this will record the data, not the sound.
2b)If you're hard pressed, and have neither, you can use a regular sound recorder, and if your piano has a Audio I/O, connect it to the microphone jack and record it as regular sound, to make a WAV/MP3 etc
2c)If you're even more hard pressed, then you'll have to consider just using a mic setup.
3) look at 2a