is it possible to play the playstation 1 games in the playstation 2?

Yes, PS1 games work perfectly on the PS2. You still need a PS1 memory card to be able to save games, you can't save PS1 games on a PS2 memory card.

Yes. The playstation 2 has full backwards compatible with the PS1. I play ps1 games on my ps2 but I think you need a ps1 memory card in order to save the game.

a region lock is say you have a usa ps2 but you get a ps1 ps2 game from japan it will not work on a usa ps2 it has a code so if you want to play them from other places you have to get a ps2 from that place and ps2 can play ps1 games

Yes it is possible. Same goes for PS2 games to PS3 games.

Yes, on ANY model of the ps2 it works better than on the ps1. (obviously)

yes but region locked

Yes, at least the older PS2's I know you can.