Hi,
Here's my two cents on your questions. :)
Housing - you should, financially, be able to find a nice place for the two of you with that much money. Availablity is another issue but I always see places in the classifieds. www.gulf-times.com or www.thepeninsulaqatar.com. If you find one that looks promising, ask us about the location.

Most compounds do have some sort of gym. Our compound has only 9 villas and we still have a very small gym. There are fitness clubs here and a brand new one (Sehati) just opened up for women only. It's a spa and gym. Also, Aspire Sports Academy also has women's only classes, very affordable.
http://www.aspire.qa/topics/index.asp?level=31&temp_id=332&temp_type=42&...
And if you can't walk outside because it's smokin' hot, the malls are HUGE.

I have an American friend (male) who lived with his Malaysian girlfriend for many months and no one bothered them. She would fly out to Bahrain every so often when her visa needed to be renewed and had no problems. Apparently, there are flights to Bahrain just for this purpose. (You have to leave the country and then re-enter and get a new stamp.) I guess she was here on a tourist visa.

Before we moved here, we contacted the Qatar Embassy in D.C. The lady there was very friendly and sent us a huge packet with brochures, a CD-Rom on Qatar and the Marhaba (Welcome) book. This stuff will give you a very basic idea of living in Doha. Marhaba is good for the "who do I call" questions and stuff like setting up phone service, getting a driver's license, etc.

Best wishes