Is it possible or practical to drive a rental car from Eilat Israel to Petra Jordan??
Technically, you can drive between the two countries. However, you have to check with the rental car company to make sure they will allow it.
Here's a website that might help with some of your questions:
Getting to Jordan
Alternatively, just cross the border and pick up a taxi from the Jordanian side. You should easily be able to get taxis to either Wadi Rum, or Petra and if that doesn't work just get into Aqaba and grab a cab from there.
Check our the LP forum (thorntree) for going rates for cabs if you choose this option. We did this in the reverse direction last year but spent a night in Wadi Rum between Petra and Eilat.