From PS2, PS3 and PS4, IF we taking about disc images, PS3 will be best choice. If from discs, then PS2. But if disc images via network specifically, I dunno how it looks like on PS3 (on PS4 not possible AFAIR).
As for PS2 gaming, the only way to play PS1 games on a network setup is using POPStarter and POPS. It's full hardware emulation, so game compatibilty is far from top notch. And it's tedious to set up.