Answers about Jordan
You can do Petra in one day ( from Amman ) , as you can spend a week in Jordan.
I will provide you with a/c car with English speaking driver, also local guide where applicable : this is the most feasible option for you as , you are traveling solo.
Prices will be function of the route ( what sites to visit ) , sure car & accommodations are a/c.
Sure, Dead Sea is a good choice, also it is safe for children; in fact they will love it , as they will float on water.
Are you visiting Petra?