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.

Here is a list of PsOne games that have several problems running on some or all PS2 models:

1. Arcade's Greatest Hits (Atari)
2. Aidoru Promotion -Suzuki Yumie- (Alyume)
3. Densha Daisuki Purareru Ga Ippai (Tomy)
4. Dragonbeat (Map Japan)
5. Fighter Maker (Agetec)
6. Formula One 99 (SCE)
7. Gallop Racer (Tecmo)
8. Gradius Deluxe Pack (Konami)
9. Judge Dredd (Activision)
10. Konya Mo Senryobako! 2000 (Hearty Robin)
11. Monkey Hero (Take-Two)
12. Monster Farm (Tecmo) - the Japanese version of Monster Rancher
13. Mortal Kombat Trilogy (Midway)
14. Namco Museum Encore (Namco)
15. Namco Museum Encore - Limited Edition (Namco)
16. Namco Museum Encore - PlayStation The Best (Namco)
17. Not Treasure Hunter (Acty Art)
18. Ongaku Tsukuuru: Kanadeeru 2 (Ascii)
19. Oukyuu No Hihou: Tension (Vap)
20. Poy Poy 2 (Konami)
21. Susume! Kaizoku (Artdink)
22. Toua Plan Shooting Battle (Banpresto)
23. Tsuiridou Keiryu Kohen (Escot)
24. Virtua Pachisuro V (Map Japan)
25. Virtua Pachisuro EX (Map Japan)
26. Wing Commander III (EA Square)
27. Metal Gear Solid (European Version: at the end is not possible to see the whole ending scene)
28. Metal Gear Solid: VR Missions

The PS2 is a 4X DVD player and it supports Dolby System 5.1 and DTS (Digital Theater Sound) sound technologies, which can be used if you have a good audio equipment and, of course, if the movie you are watching supports them. A Sony remote control to use the PlayStation 2 exactly as a DVD player is sold separetely, and there are also many good third-party remote controls.
The PS2 is a great CD player, so also your Audio CDs will take advantage of the DTS and the Dolby System.
But what is even more important is that PlayStation 2 games come usually on DVD; a normal DVD can store 4.7GB of data, against the 650MB of a normal CD - a support with high storage capacity is fundamental for newer videogames that can feature movie-like sound quality and high-definition textures.