Jordan is a really nice country to visit, and it's still very safe for all tourists. The people are welcoming and very helpful.
And all the places mentioned in the question are really safe and no risk factors, do not drop any place they are all really a should visit places.
You can visit Petra and Wadi rum together since they are on the same way and you can continue to Aqaba (The Red Sea), spending a night at Wadi Rum is a must, to watch the sunset and sunrise over the mountains there.
Dead Sea and mount Nebo are on the same way too, you can go and spend a night at the dead Sea and next day go to mount Nebo, also I recommend to go Ma'in hot springs also called Hamamat Ma'in in Arabic, also there is Wadi AL Mujeb there to go hiking through water.
Come visit Jordan with no hesitation. It will be a trip to remember.