Off Dubai, you have the wrecks, unfortunately the Palm Island (with all the dredging) has mucked up the vis and changed some of the currents, but it's still ok. To the north, you have Ajman where you can dive with manatees and dolphins..
Go to Musandam and do all kinds of dives (not that many wrecks though), wall diving, coral reefs, drift dives etc... Dec - March, you have a chance to dive with whale sharks, reef- bull- hammerhead sharks, dolphins and so on...
Oman: Dimaniyat Islands, go to Al Suwaidi and you have a string of 9 islands with about 20-25 different dive sites, loads of fish, corals. The Islands are a marine park and protected by law.
I do most of my diving there and really recommend it. If you want more information, send me a private message and I can answer any and all questions you have.
For the Red Sea, well, I'm not the expert. I know that some people I have talked to said that, even though the vis is better in the Red Sea, fish life is about the same as off the Dim Islands.
In Oman you can also head down south to Turtle Island (they might have changed the rules there, but you used to be able to dive there), and there is also diving off Salalah, but I have never been down there so can't tell you...
Temp's, well you live in Dubai so you know what it's like. Humidity is a bit worse here in Oman though...
Best times to dive? Anytime. visibility is improving at the moment, still good to dive though...
At the moment in Oman, I'm using a 2/3mm full, but in a couple of weeks, a shorty should be alright, come late-April and May, a diveskin should be alright -- all of it depending on how much you dive. Some use a 5mm full, depends on how easily you get cold...