Answers to your questions:
School--check the website for the American School of Doha
It is an accredited school with both American (AP)system and next year will begin the IB system as well. QA should pay the full costs of tuition. Be aware, however, that there may be a waiting list. Apply now.

Music-- check the International School of Music. they offer top notch teachers for both piano and violin. My kids have taken both there. The cost is about $30 for a half hour lesson. They have recently started 1 or 2 other music schools. I am not familar with those.

Stay at home moms-- With a full-time or part-time maid your wife will have more leisure time. There is an American Women's Association, many charity groups, A potluck group, tennis lessons, cooking lessons, everything you could think of! She will only be bored if she lets herself be. I find myself socially busier in Doha than I ever was in the states.

Cost of Living---check previous threads. Gas is 80cents (yes cents) a gallon here. Everything else is more expensive.

Diving-- several clubs available. Also check threads.
Good Luck. Adventure awaits you. Doha is far from perfect, lots of irritations and frustrations, but a great learning experience.