Yes, sorry, the above are all advertising posts!
Obviously if you are going to be needing a car and driver it is likely to come out expensive. Presumably you have already realised that. Putting aside the transport question for the moment, here are some observations on these sites for the disabled.
Petra : the main site is some 2 kms from the entrance gate. There are small carriages, about a hundred meters down from the ticket office which will take you to the Khazneh (the Treasure). The standard charge is 20JD for 2 people which includes the return, the driver will ask what time you want him to be waiting for you, and they respect this time pretty well.
From there down to the main site, you can have something like 2 hours normal walk and back to the Basin, not stopping for anything very much. Almost certainly this would be too much for you. And of course one DOES want to stop along the way!
You can hire a camel which is fine for the flat, but a donkey would take you further and more easily - Petra has a lot of ups and downs in it. I recommend you very highly indeed to do this. You are probably looking at around 20JD for the day. You will have to manage the crutches as best you can, since there are some places where you will need them. Mostly these donkey boys are helpful, conscientious and sympathetic.
Wadi Rum : as you say, here you can hire a guide and a 4x4 for your visit. If you do not organise this in advance, you will have to take the next guide on the roster at the Visitors' Centre. The problem is that these are really "drivers" not guides and many of them do not speak English. I heard from somebody who complained that the only English his "guide" knew was "Big Rock" which he offered, perfectly accurately, every time they passed one! It's more difficult than it sounds to "organise one from Aqaba"
Madaba : you are looking to visit a small town. The main sites, the museum, the Church of the Map, etc are all easily accessible, with not very much walking.
Mount Nebo and the Dead Sea Panorama the same. Mount Nebo does have something like half a mile or more of walking to get to the view point from the road. The Dead Sea Panorama rather less. But they both have steps to negotiate.
Um ar Rasas, much the same, as long as you don't try to explore the "new site" which has recently been opened. The mosaics in St Stephen's Church are easily viewed.
Wadi Mujib : sorry, but forget it! This is one of the greatest adventure tours in Jordan and involves hiking over rough ground for the very easiest tours; the gorge itself includes abseiling in a number of places and scrambling over rocks in water up to your waist or deeper. In the spring you might have to swim. It's deep in a canyon and there is no way to see anything of it without a lot of walking/scrambling.
Try visiting Dana instead! You have a magnificent panorama easily viewed from the Resthouse terrace and a reasonably easy walk around the village on the Tribal Arts trail.
Kerak and Shobuk : well they are castles and do need a fair bit of walking around, and a lot of step climbing, but I expect you realise this.
I do suggest that you try again to contact a few agencies who can organise this all the way for you, including the Wadi Rum part of your stay, and who might possibly get permission for a car to take you closer to the sites like Mount Nebo than is usually allowed. It is also sometimes possible to get permission for a car to collect you from down in Petra so that you wouldn't have to worry about coming back again after the trip down to the Basin.
Try emailing one or more of the following. They are small agencies who can give you individual attention, but they are all efficient and experienced. I should be very surprised if they do not reply!
Petra Moon Travel Agency in Wadi Mousa. (This agency also offers hiking and camping trips) Tel 962 3 215 6665 Fax 962 3 215 6666
Website http://www.petramoon.com email info@petramoon.com
Nyazi Tours in Aqaba. Tel 962.3.202.2801 Fax 962.3.201.9461
Website http://www.nyazi.com.jo email nyazi@index.com.jo
Jordan Inspiration Tours www.jitours.com. This is owned and managed by Sami Hassanat in Petra, who is a qualified travel guide. His email : sami@jitours.com, fax 962.3.215.7317 mobile number 962.745.554.677.
Good luck!!!