Here's som example rental agency websites for some pics, etc. These sites were helpful to me before moving here.

http://www.zukhrof.com/

http://www.homes2rent.biz/Serviced%20Apartments.htm

http://www.mirageproperty.com/

I think a FF 1-bed+hall would be well within your budget.

Work visas are sponsored by the employer who brings you in, so yes, you need to have a NOC (no objection certificate) or sponsorship transfer granted by current employer to change employers. Any reputable employer, especially companies not based in Qatar, always issue these, as long as the employee has performed well.

Here's some more guides with general info for you:

http://hr.web.cmu.edu/forms/benefits/qatar/practicalguide.pdf

http://www.qp.com.qa/qp.nsf/web/living?OpenDocument