With regard to your road trip from Damascus to Amman, indeed it is very much possible (and the preferred way for many travellers). With the recent terrorist attacks in Amman; however, I am not sure if land boarders are open/easy to get through to be very honest. As GK mentioned, check with the hotel upon arrival in Damascus and take it from there.

Petra is very hectic for a one day trip. Usually folks leave to Petra the day before, spend the night in a nearby city and head to Petra in the early morning hours the next day. Just an FYI...