Does PCSX2 play PlayStation 1 (PSOne / PSX / PS1) Games?

No, simple as that. PCSX2's primary goal is to emulate the PS2, which would eventually include PlayStation 1 games, however at this time the key focus is to make PlayStation 2 games 'run' with a high degree of compatibility, including support for such features as USB device support (EyeToy, Special Controllers), DEV9 (Network, HDD) etc.

If you wish to play PlayStation 1 games, there are extremely compatible, fast and stable emulators already in existence, and I recommend:

pSX - New emulator, with a growing level of compatibility and is very simple to use.
ePSXe - The famous and widely supported PS1 emulator, use this if you love features, and plugins!
PSXeven - Created by Xeven, supports the PSEmu Pro plugin system, and often has increased compatibility over ePSXe with some specific games.