I travelled through those two countries just a couple days ago, although only to Amman in Jordan (not Petra). I stayed at the Le Meridien downtown, which is a great location. The airport express bus stops right behind it and is much cheaper than a taxi (fixed rate 19 JOD).

1. Jordan is very modern. The taxi drivers were very professional and friendly, even the non airport ones (or perhaps, especially the non airport ones ). Cairo/Alexandria are very backwards compared to Amman.

2. Yes there is a crossing into Israel from Jordan. Be sure to have Israeli visas in hand before you cross.

3. No problems with passports.