Agree with the tight traffic in Amman but I rented a small Kia Picanto, pretty similar to the size of car I drive in the UK. Hardest thing was learning to drive with a gear stick on the other side, and a clutch that is so low (mine is on its way out)

I was advised to only drive in and out of Amman by other travelers but found it no where near as headache as they made it out to be. Even got caught in rush hour on my back, just reminded me of UK stop start traffic.

Also the full coverage for my car was pretty reasonable so got it, though didn't need it.

Traveled there last year and met a few women traveling, either with friends male and female or solo. They had no issues, and most of the time didn't have to cover up. Saw plenty of tourists in shorts and shoulders showing. Depending on where you go maybe be a little more mindful of what your wareing but tourists in Petra, Aquaba, Jerash and wadi rum where dressed just like any tourist spot I've been to in the world.

Literally did everywhere you listed, tho could go back. No photos online to the immense scale of Petra any justice, I walked around the site for 8+ hours and could have done another whole Day

The most friendly locals of anywhere I've traveled (30+ countries). Great weather, incredible history, good food and super easy to get around. I rented a car and it wasn't to expensive and the roads where better then at home (UK) tho the is a lack of signs for speed bumps, that are just black lumps of tarmac.

Ps. Did see a local on a sport bike on the outskirts of Amman when coming back from Jerash. Tho didn't see any rental motorbikes