We hired a car with driver during our visit to Jordan last April and I must say it was a great experience. My boyfriend refused to drive in Jordan after his friends scared him off, saying how easy it is to get lost and how difficult to drive after dark. We've met a lot of Westerners driving there with no problems but I must say being with an experienced driver who knew his way added to the relaxing and pleasant experience. We must have been lucky as he never tried to rip us off and even negotiated better rates for us for water, fruit etc. I have checked the 'going rates' for tourists and he always charged us a little bit less.
So if you want I can send you his details.
We've used him all the way to Dana, as from Dana we had pre-arranged transport.
In Petra you don't need a car and from Petra to Wadi Rum is 40 JOD max. You can haggle for less.
From the airport you take a fixed price taxi so you won't get ripped off. It should be some 35 JOD so not much more than your rental car.
After past experience I would never go for the cheapest car rental (unless recommended by several travellers) as I've had a hair rising journey through Turkish mountain in a cheap car, which literally started falling apart- a good story but not an experience I would like to repeat. You need a car you can trust your life with!
We arranged a local Bedouin to take us from Feynan to Dana via a little known route through Little Petra. This is not a route you can do on your own and even with an experienced local driver it was not for the faint-hearted. So beautiful though that it's one of my most fond memories of Jordan. It cost 40 JOD on a beat-up pick up truck.
Wadi Mujib Chalets to Kerak is not such a long journey, are you going to stay overnight in Kerak?
We went from the Dead sea all the way to Dana village, via the Kings Highway and with 2 hour stop at Kerak and a stop at Wadi Mujib and Tafila mountains and paid 60 JOD.
I will leave doing all the maths to you.