In addition to and support of Deano's answer:

1 - Is it possible to take an ps1 or ps2 from europe (PAL) to USA (ntsc) and use it there with a transcoder or something like that?

YES - WILL NEED A 240/115-50hZ/60hZ MAINS TRANSFORMER

2 - Is it possible to use the game enhacer cd in a ps2 to play ps1 backup cds?

DEANO'S ANSWER APPLIES


3 - Is there any game enhacer for ps2 to play ps2 backup cds?

DEANO'S ANSWER APPLIES

4 - I'm willing to buy a ps2 but I would like to play ps1 backup cds (cdr) on it so I ask for anybody a recommendation if I should buy a ps1 or ps2?

AS DEANO SAYS BUY A PS2. YOU'LL GET BETTER PS1 GRAPHICS

Taking a PS2 to the USA probably invalidates the warranty because of the awful cartel that localises PS2 to a regional basis. So you might as well have it chipped. If you're going to the US soon and you're going to stay there a long time, why not buy the thing there?