To answer question 1, I think it depends on how many trips you want to go on in a year and how much vacation time do you have. DXBPlat has listed a lot of good places which are close but if you're only going for 1-2 trips per year, maybe you'd be willing to go further and spend a week or more.
One of the best things I've found since moving to the middle east from the US is the number of interesting places within a 6-7 hour flight. I would concentrate on these as you're in a central location and can get to places you'd never fly to from the states. As mentioned previously, the biggest part is getting out of Saudi to Qatar, Dubai or Abu Dhabi to connect further.
If you want to see different cultures then you've got all of India to explore or go south to Kenya for a safari trip or South Africa; all very far from the US but not that bad to get to from Saudi. In SE Asia, you've got Malaysia, Thailand, Singapore and Indonesia and you could do a single trip that connects a couple of these, we did Singapore-Bali for a city-beach vacation last year.
If you decide to use Dubai as your connection point, think about staying here for a couple of days to see it (and shop) before connecting on Emirates or the budget alternative FlyDubai.
For amazing water and beaches, Maldives can't be beat (but very expensive) and you've also got Seychelles and Mauritius as 2 slightly cheaper alternatives.
For #2, I would get a local credit card and use it instead, Amex is not always accepted everywhere. Pick an airline that you're flying with and see if they have a credit card that offers miles. Emirates airlines (one of the best, based out of Dubai) offer a Citigroup and Emirates Islamic Bank credit card. This will help you earn lots of miles and free future tickets, especially if you book your tickets with the card as well.
In summary, pick a major airline that you prefer (Emirates, Eithad, Qatar airways), check out their route map and enjoy
