I mean, I wouldn't expect them to since PCSX plays PSX games fine enough on its own.

PCSX2 is basically the PS2 if they removed the backwards compatibility...you can play PS2 games nicely on it, but it doesn't support the previous generation of PSX games. You'll want the first one, PCSX for that.

 

Definite no.

Use a ps1 emulator for ps1 games and PCSX2 for ps2 games.

 

Well sorta, pcsx2 can run PS1 games but very poorly if at all, it's something the devs tried but quicky gave up in order to concentrate on PS2 emulation.
You can even use pcsx2 gdsx plugin with a PS1 emulator like pcsx or ePSXe, but again I wouldn't recommend it.
Both pcsx-r and ePSXe will do far better job with thier own plugins.