You can use POPStarter to use the built-in software emulator, but it's not great. There's also FreeMCBoot and MechaPwn if burning discs isn't a problem (might only work on some models? I'm not sure)

I modded my PS Mini for PS1 games. Its pretty simple.

What ShinkanTrain said. The last a read about it, the PS2 only switches into PS1 mode on a trigger from the optical drive subsystem, and then most of the memory and other hardware used to run homebrew is deactivated. AFAIK no-one's yet found a way to trigger the change in software and keep the connection to wherever you're loading your game from.

I believe that on certain revisions of the console, MechaPwn can overcome the protection, but you still need a "Playstation 1" CD in the drive to actually run something, as ShinkanTrain wrote.

Probably because it's pretty slow, and the custom drive format used by the PS2 isn't very flexible; game images have to be in one continuous block, and blocks can't be moved. You can overwrite one game with another, but only if it's the same size or smaller. So if you delete games off in the reverse of the order you put them on you're fine, but otherwise you're going to leave empty "holes" of wasted space.