That sounds like such an amazing adventure! I have only been to Dubai, out of the places on your itinerary, so I only have recommendations for there.
Gold Souk - amazing!
Lunch or dinner at the top of the Burj Khalifa (they had a special that was something like $50/pp for a prix fixe lunch, that was the cheapest way to get to the top of the world's tallest building). The dancing fountains outside of the Burj Khalifa are even bigger and more amazing than the Bellagio in Las Vegas.
If you're really into drinking/eating, I heard from a British expat I met there that they have these crazy weekend brunches, all you can eat and drink for hours, in the $100+ range.
The restaurants are truly amazing, you sort of can't go wrong.
If you a motorsports person, I can highly recommend the dirtbike tours - they'll do a half or full day, set you up with full riding gear and a dirtbike specially equipped for the sand, and take you on a guided tour through the sand dunes, where you will see camels and gazelles.
the mall! I'm not much of a shopper myself, but it is amazing to see the mix of people, the luxury brands, and the strict enforcement of the no-PDA rule (not even hand-holding). It is worth doing a bit of reading up on cultural expectations, as you can get thrown in jail for public displays of affection or public drunkenness.
There is only one mosque that a non-Muslim can enter, and they do offer tours/brief lectures which is worth going to if you are curious about that.
Take the metro someplace! It's not always the most convenient option, but it is literally the most clean form of public transportation you will ever see.
I'm sure there's a million things more to do, have fun!