Yes it can be chipped. All versions of ps2 can be chipped.
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The PS2 Slim Plays all the PS2 games the Fat PS2 would.
They are designed for a Fat PS2 and need an adaptor to fit the slim models one amazon customer said "The second problem came to light when I tried to plug in 2 of these things; the receiver units don't fit next to each other in a slimline PS/2 very well; they're just a little too wide, and the receiver cases bump into the PS2 case at the top, preventing them from being plugged in the whole way. I ended up putting both units on my belt sander to grind them a little narrower, and I ground off a little of the back at the top to get them to plug in the whole way. Maybe they fit fine in the regular PS/2, but definitely not designed for the slimline version." see related link
The PS2 plays games on discs and burnt games do not work with a the PS2
the larger ps2 you can ...yes not the slim ps2
No it is not a PS2 game so it will not work on the PS2