MY PS2 is also like this, it does the same things you are discribing but mine is picky and only does it with select few games. I opted to just get my good old PSone out and continue to play PSone games on a PSone. I didn't want to hurt my PS2 or games.....maybe if you can, you'll want to do that? :fd:
(I bought my PS2 a long time AFTER they were released, so I don't think this has to do with any bugs that accour in systems during the first few weeks-months of release.)