Hello - yes it is possible for Aqaba to Petra and you must pay the taxi driver when you get back to the ship. Likewise it is possible to get a taxi from port Sokhana to Cairo and again you will pay when the taxi driver drops you off back at the port entrance. Cannot comment on Luxor except it is a long way by taxi - again it may be possible but never did that myself.
If the ship arrives in Aqaba the day before you go to Petra - I think it is there two days - you should go to the port gate on the first day and make some arrangements with the taxi drivers who will then be there the next day so you are guaranteed a taxi and return trip - that's what we did and had no problems at all - and it was much quicker than the coach due to the windy steep roads.
However you cannot do the Dead Sea and Petra in a day - so if you want to do both then you will have to do one on the first day and the other on the second - I think you would be best doing Dead Sea on the first day as there is less to see and you will have less time due to arriving in port at 11.00 am approx.
The 2 in 1 day looked feasible on the map, and it seemed a nonsense to do a big chunk of the same route twice, but Mr Google's never been to Jordan (or Egypt) so couldn't give me time or distance, and I thought the desert was flat!!! So we'll go for just Petra, skip Dead Sea (by itself, not worth the long trip from Aqaba) & mebbe on the short day do Thomson's Wadi Rum trip. Because of the distance & amount to see, we reckon to do Luxor on Thomson's 2-day trip. Thanks for your help. Chris.
If you arrange to leave by taxi at the crack of dawn then you will be able to do both but you will have to get your skates on ! Go to Dead Sea first and then call at Petra on the way back - I agree about the chunk of the same route - I would suggest that as the ship arrives at 11.00am on the first day in Jordan you go to the port entrance and arrange to meet a taxi the following morning at say 6.00 am ( get an early breakfast in the Lido ) - you would be at the Dead Sea by about 9.00 am - have an hour or so there - then get to Petra by 12 noon then 4 hours there and leave at 4.00 pm and back at the ship for 6.00 pm to sail at 8.00 pm - it's a rush but just about possible. I think you need 4 hours at Petra if you are fit and active - to speed things up there get a horse taxi down to the sights rather than walking and then 4 hours really is more than enough.