If you look at other posts, you'll see that the accomodation situation here is bad, and it would be difficult for you to find anything affordable just now. Things are expected to improve after the Asian Games, however, so it might be an idea to put up with the offered shared acco for a month, and then see what's available in the New Year.
The Asian Games will also hamper your forays into society. The traffic's gonna be horrendous and some clubs (eg QNHG and Doha Players) won't even be trying to have functions. However, you'll get ideas from Marhaba magazine (available most supermarkets and hotels). If you're into bar culture and feeling bold, visit big hotels and Garveys or the Rugby Club.