While completely illegal, Dubai is a mini-metropolis and all activities take place. But I wouldn't sit down in a park and toke up if I were you.
There's very little fun harassment. But my fiancee visited a few months ago and when she went to the shops she was wearing a strap-top, and one guy crashed his pushbike into the back of a police car whilst looking at her. She found it amusing, not threatening. There's very little actual assault here.
Check out any of the maps for the location of Satwa. It is the area NW of interchange 2. Doesn't have the highest reputation in town but ok. Good place for room-renting. If you get there check the noticeboard in Lal's supermarket (open 24 hours) for accommodation.
Ramadan isn't exactly 'celebrated' - it is 'observed'.
It is Dubai law that nobody can eat or drink in public during daylight hours. Although I haven't heard of anybody actually getting arrested in Dubai, the police in Sharjah and Abu Dhabi have been known to take a harder line. 'Be respectful' is the motto. Oh - and although most bars are open (if they're not closed for refurbishment) drinks are only served after 7pm, and there's no music.
In Dubai you can _buy_ food at most normal outlets (in Qatar you can't buy anything anywhere). I guess Sharjah & Abu Dhabi are a little stricter? Just take your food away from the petrol station in a plastic bag. Don't sit in your car at the edge of the road and eat a sandwich - get back to the office/home.
Actually it is no hassle really (not here in Dubai)
Oh - and it probably isn't the best month of the year to stroll round near mosques with a mini-skirt and string T-shirt
HTH - S