Newbee, Doha is a fairly normal city, I would compare it to a midsize US City. If you are coming and have to pay your own rent, you will have problems, $1,500 to $2,000 per month for Apt, most you wouldn't want. COmpound is like a gated community, most have pools, etc. Outside is a house on a street, etc. Safty is same, no problem, crime, no problem, terrorist, no more problem than other safe places. Safer here than the US.
Women need to dress mostly normal, don't cover heads, sleeves are good, shorts normally at beach or clubs. Lots of capri pants. Women can do most anything normal, they drive, they work, it is normal.
School cost can be a problem, Amer School of DOha is hard to get into. Ask your employer for education allowance. K-12 schools for americans.
Health care is coming, lots of people have kids here. Check out

http://www.expatmum.com/vbulletin/forumdisplay.php?s=&forumid=16

There are 5 american Universities here so there is a fair American group of people. 7,000+/-
All the Universities have information on their websites for their staff moving to Qatar, good info.