Here in the UK, I can hire a plane of any registration and fly with my FAA certificate. I still have to get checked out by the owner/club, but do not require a conversion or type-specific rating.
I believe (have not confirmed) that I can fly a G-reg (UK) or N-reg plane to the EU with my FAA certificate also. I don't know if I can fly 'from' the EU with my FAA cert, I haven't asked (as in hire in France or Germany).
In Botswana, Namibia and South Africa (when I was there about 2 years ago) a conversion is required for non-N-reg aircraft. Additionally there, one needed to be rated in the specific make/model.
as for travelling in an N-reg plane to other countries, it's the same as if you wanted to fly to Canada or Mexico from the US. you have to fill out a bunch of paperwork (EU inter-country flights are not quite so bad) but you can generally fly an N-reg plane to any country.
There are some issues, especially since you're in Israel... ie. you can't go to Saudi Arabia...