Some comments:
- I'd definitely add in a Baalbek visit to your Beirut itinerary. Roman ruins in great shape with almost no tourists; unless you're bored out of your mind by Roman ruins (NTTAWWT), it's a can't miss. It would work well for day 4. One full day seeing Beirut city cites is fine.
- There are a few easy and common tourist stops on the way to Petra via the King's Highway; if you already have a car, you might as well hit some of them. The Madaba map is pretty cool and can be seen quickly. Mt. Nebo (where Moses died) has amazing views and is very important from a biblical history perspective. Kerak castle is okay.
- Amman just isn't that interesting. Jerash, the Roman ruins near the city, is a half-day activity; nothing much in Amman is very interesting. Two full days in Amman sounds very excessive.
- Definitely do an overnight with the bedouins at Wadi Rum. Many different options depending on your level of luxury -- 5 star camping to roughing it. You can park your non-4x4 rental at the entrance to the village, and your bedouin host will pick you up in the 4x4.
- The Dead Sea in Jordan isn't exactly a tourist cesspool. Way more tourists on the Israeli side. I think the Dead Sea is cool and I've been about 4-5 times total on both the Israeli and Jordanian side, but it's a quick experience -- go in the water and float and take a pic, and maybe put on the mud if you're into that. People make it into a resort experience, but it's not my thing and it doesn't sound like your thing either. So visit it quickly and move on.
- A day trip to Jerusalem from Amman is logistically impossible for a variety of reasons. Consider doing an open-jaw ticket and flying out of TLV