Let me answer Q1 as a start.
Your biggest issue in the Gulf is the general lack of low-cost carriers to let you get out of Saudi Arabia, there are FlyDubai and AirArabia from the UAE which do help but generally from Jeddah you're looking a good $400 a head just to get out to a hub in Dubai, Qatar or Abu Dhabi. Welcome to the world of the non-free market.
Anyway, assuming you can find some sort of deal to get yourself out with a good connection, the world is your oyster here. My recommendations would be:
Oman: Muscat is beautiful, full of culture and nice hotels, the nicest people in the Middle East too
Beirut: A modern, lively city which feels like the south of France. Lots of old Roman ruins within an hour of the city, skiing too in the winter.
Cairo: Present situation aside, again full of culture (pyramids, city of Saladin, museum) and as busy and vibrant a city as you'll ever go to. Expect to get ripped off as a Westerner because you haggle for everything, but the good news is that Egypt is so cheap that being ripped off is pretty much normal tourist prices anywhere else. Sharm el Sheikh is also an OK beach resort if you want that.
Jordan: A long weekend taking in Amman then the Petra ruins is good. Alternatively, instead of Amman go to the beach resorts in Aqaba which are closer to Petra.
In terms of Africa, Dar es Salaam (Zanzibar) and Kenya (safari) aren't far away, but not that cheap to get to.
For Asia, Thailand, Malaysia are the cheapest places to get to. India is also near but as so much travel for workers to the region it's not as cheap as you'd expect.
In essence, stick a pin in the map and go, largely you can't go wrong. In terms of absolute places to avoid, Iran and Iraq are obvious ones! Given topics on here, many Americans want to go to Israel - my simple advice is do not even think about it. If the KSA government or your employer finds out your new life in the sun will come to a very abrupt end. It's not just the issue of the stamp in the passport, as a Western ally KSA is tapped in to all the international databases and will know if you go there. It is not worth the risk.