what is the difference between forge and fake as verbs and which correct to say "he has faked the passport" or "he has forged the passport"? and which correct to say "fake passport" or "forged passport"? and if both are correct then when to use each of them.
Ignoring the fact that the word "forge" has an alternate definition in the field of metalworking, the difference between the two primarily relates to the object of the faking/forging.
In your usage above, the two words are synonymous. Both mean that he has produced an inauthentic passport.
However - you would only use the word "forge" in relation to written documents and signatures, whereas you can use the word "fake" in many more contexts. You can fake an illness, pretending to be sick. You can fake your own death to deceive others.
Kevin makes some good points in his answer, but there is an additional distinction. Something that is forged is supposed to appear to be the real thing. It is intended to deceive. Something that is faked may not be.
When someone is driving a car in an old movie, there is often a changing background behind them to make it look like they are driving. This background is faked, but no one would think that it was a real background or that the movie makers were trying to deceive their audience. It's just faked to help your suspension of disbelief.
This is also true of the adjective versions of these words. A fake passport might be a prop used in a play or game, which couldn't possibly stand up to inspection by an actual customs agent. A forged passport is intended to be used in place of a real passport, and if the customs agent realizes it is forged then it is a poor forgery.