I have spent more than 4 years in Dubai and have never been cheated by a cab. There is a meter which is always used without asking, 20 AED flagfall for airport pickup, elsewhere I think 3.50, but a 10 AED minimum (which effectively means a 10 AED flagfall). Add an extra dirham for call-to-pickup.
It really doesn't matter if you are male or female either and Dubai is a good introduction to the Mid East. Oman is a must and I'd second the Beirut idea. You can skip Doha, but a day in Kuwait would be nice if it is convenient with flights (say to/from Beirut with a Kuwait stop).
If you want something really different find an agency in Yemen (I can suggest one) and book for perhaps 4 days in and around Sanaa. You'll have to see what the political situation is in Yemen when the time comes, but I can't think of a more interesting city in the world than Sanaa. Air Arabia (excellent) has cheap flights from Sharjah to Sanaa (and all over the region too).