Any of your options would work with no problem.

You can easily take a taxi from the airport directly to the Dead Sea (if you don't arrange it from your hotel), for maybe $40. I would expect to spend about JOD 100 ($140) or less per day for all-day car and driver hire from an agency. If you want to do it cheaper, you can ask pretty much any taxi driver (e.g., the one who takes you from the airport) and he will make either himself or his brother/cousin/friend/etc. available to meet all of your transportation needs. I have done this many times to the Dead Sea, Petra and Wadi Rum, and Damascus, with good success -- the drivers are friendly and honest and speak at least a bit of English.

The taxis won't be an luxurious as a car from an agency, of course, and maybe not advisable for the longer trip to Petra. You could also rent a car and drive yourself for slightly less.