We visited all those places when we were in Jordan in 2006 - our photos and travelogue can be found at www.geocities.com/PaulKilfoil/2008.htm</a>.
We booked our accommodation ahead, often by email while on the road, 2 or 3 days ahead.
King's Highway : This is not a well-travelled public transport route - most buses go on the fast, boring desert highway. However, the Mariam Hotel in Madaba (an hour from Amman) organize regular buses from Madaba to Wadi Musa (Petra) via the King's Highway, leaving from right outside the hotel. This bus also stops in Karak for a couple of hours ; Karak Castle is worth wandering around in, it's an impressive place. We did this trip and it was great. The bus dropped us at our hotel in Wadi Musa. The trip from Madaba to Wadi Musa cost 18 JD per person. The Mariam Hotel is a bit of legend in Jordan, certainly the best place we stayed in anywhere the Middle East and not even expensive. Their web site is at www.mariamhotel.com</a>, email reservations@mariamhotel.com.
Dead Sea : Best way to do this is to organize a day tour from either Amman or Madaba. We hired a car and driver for the day via Mariam Hotel and it worked out well - we went to Mount Nebo, the Dead Sea and Bethany (where Jesus was baptised by John the Baptist). Very hard to do this by public transport. There's not much at Mount Nebo but Bethany and the Dead Sea are definitely worth it.
Jerash : Only a couple of places to stay here. Hadrian's Gate Hotel is cheap and convenient - it's directly opposite Hadrian's Gate. Their phone number is +962 7 7779 3907. The Olive Branch Hotel is a nice place but a long way from town in the hills above Jerash.
Wadi Musa (Petra) : The hotels down at the Petra gate are all over-priced and not great (except for the Sunset Hotel, which is cheap and not great). A far better idea is to stay somewhere in town (Wadi Musa) and take a taxi down to Petra every day - taxis ply this route non-stop and are cheap, 2 JD per trip. It will cost way less staying in Wadi Musa and using taxis than staying down at the gate. In any case the walk down to Petra from Wadi Musa is easy - it's the walk back UP the hill after a long day exploring Petra that is deadly.
Wadi Rum : Good idea to book beforehand. You can arrange stuff when you get there, but there is a vast difference in quality between some of the dsert camps and others. We used Atayuk Ali and were very happy ; their desert camp even had showers!