No, PS2's can not play music while you're playing a game on it. In fact, the only game console that can play music (for certain games) while you're playing anything in the PlayStation family, is the PS3.
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No Nintendo DS games were not designed to play on the PS2 and the PS2 can not play them
Crysis is a PC game and Crysis is a PS3 game and neither can play on a PS2. Look for PS2 games and then you will know they will play on a PS2. There are thousands of them for you to choose from
Yes
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No you can only play PS2 or PS1 games on your PS2. You can also play DVDs and CDs