Just A Couple of Days in Jordan ...

Sure. There's no reason not to visit Petra and three days is sufficient.

You have a healthy budget and I'm going to suggest you do it a bit different. Renting a car and driving yourself is fine, but you're stuck paying for a car you don't need for a couple days. It wouldn't be any more expensive to simply take a taxi from the airport. The taxi might seem expensive, but when you factor in the price of the car you're not using while in Wadi Musa (Petra), it won't be any more expensive.

There are no districts in Wadi Musa. It's too small. On your budget there are several high-end hotels to choose from. My advice is for you to check TripAdvisor. There are a couple high-end hotels right at the entrance to Petra and places like the Marriott which are a little further away.

There's certainly no reason to be nervous about anything. Jordan is very safe, the trip you propose is easy to do and Petra is worth the visit.

Good luck.

Thanks for the info. Your suggestion would be for me to stay near Petra rather than in Amman?

If you're going to visit Petra then you need to stay in Wadi Musa. It's a three to three and half hour drive each way to Amman. Staying in Amman wouldn't make any sense.

KoalasRule - with your budget I would be looking at a 'stopover' package with the airline you are flying with. It is called a stopover package however they are used by anybody who wants to have a side trip discounted by the airline.
Have you booked your airfare yet? If not, the whole deal gets wrapped into your air ticket price and you just pay a tip to the driver who picks you up at the airport and the hotel for whatever meals are not included in the hotel price. Go back to your travel agent (assuming you have used a human being!) and discuss this with them.
Alternatively, contact the airline you are flying with and ask them some questions for a guide to prices.
Then compare with a tour company for example Intrepid http://www.intrepidtravel.com/trips/EEAV-Ohas a 3 day trip which includes Petra and the Dead Sea.
Adventure World at http://www.adventureworld.com.au/tours-and-destinations/asia/middle-east/jordan/tours/essential-petra/ has a 3 day tour of Petra.
Thre are several other companies that do something similar with hotel, airport transfers and entrance price to Petra included.
Another company that I use in Sydney when I want to look at a brochure and compare costs is The Adventure Specialists http://theadventurespecialists.com.au/contact_us.html

Thank you for the tip. However, I purchased my airfare some time ago, so it's not possible to add on a stopover. I've booked, planned and paid for pretty much everything but now find that I have a few spare days and thought it was too good an opportunity to miss.

Would recommend you rent a car though. Driving is easy and it will enable you to visit some other places as well if time allows. Might want to make Madaba your home base for the first night.

I wouldn't be nervous about driving in Jordan except in Amman.
If you make up your mind to stay overnight in Madaba, which is a good suggestion IMO, ask on the forum how to find the road to Madaba from the airport. It seems that several persons have had difficulty with the road signs.
Enjoy,- you will have a great time !

I traveled alone in Jordon, Amman to Petra to Aquaba a few years back as a single woman in her 50's.

no problems, very safe. I didn't rent a car, but driving outside of amman seemed fine.

do you like snorkel or dive ? aquaba has amazing reefs.

at petra get there first thing in the morning, no one else around....very amazing colors and light.

Have a wonderful trip.