Aqaba to Eilat taxi?
 

How long is the ride, and how much time should be expected for border formalities going each way? I was planning on going from Aqaba to Eilat for a few nights, then returning and taking a bus or flying from Aqaba to Amman before making my connection back to Paris the next day.

It may take longer Jordan/Israel than the other way around. In any event, it's time of day that will make or break the crossing. We crossed from Jordan to Israel at 5:00 (1700). Took a couple of minutes. I was told that if you arrive a couple of hours later, when all the Petra day-trippers return to Eilat, it will take longer.

I hope you realize that you can NOT take one taxi from Aqaba to Eilat. Neither are allowed to cross the border.

Correct. Take a taxi to the border. You then walk across the border and keep your fingers crossed that a taxi is dropping someone off. ;)