General Information - Compatibility
Yes to both questions. The PS2 has a "100% backward compatibility", according to Sony, which means you can play older PlayStation (also known as PsOne) games on the PlayStation 2. The PlayStation 2's texture interpolation option can also make the graphics of some PlayStation games sligthly smoother. Many PlayStation games, listed below, do not work perfectly when played on the PlayStation 2.
You can use all your PsOne peripherals on your PS2 unless you are playing games which require the Dual Shock 2 or other full analog controllers. For example, you cannot use your old controllers with Gran Turismo 3. This game, in fact, uses the Dual Shock 2's pressure sensitive analog buttons. Also, memory cards for PsOne can be used only to save PsOne games; it's also not possible to save old PsOne games on PlayStation 2 8MB memory cards. If you want to save a PsOne game on PlayStation 2, you need to use a PsOne memory card; if you need to save a PlayStation 2 game, you need a PlayStation 2 memory card.