Generally speaking, what would it be like for an American woman living in Qatar?

It would be great! It's safe and beautiful and as long as you respect the culture there, you'll be treated like a guest. You'll find more foreigners there than Qataris themselves- though if it's your first visit, you'll take time to figure out who's who. There are a lot of Arabs from both Asia and Africa other Asians like Chinese and Indians and Filipinos, etc, as well as other westerners, and all kinds of people you'd not expect to find there. It would be a wonderful and rare experience for you! Make sure you know all about the place before you land there, though. It would be a good idea to read a few books as well, as they will give you a complete and more reliable over-all picture than any article out there. If you're planning on living there, then you must devout a good amount of time and money to learn about the place you're going to to. As for crime, the crime rate in Qatar is relatively low compared to industrialized nations. Stuff like pickpocketing and bag snatching does occur, but is extremely uncommon.