Ask about how long it will take to get your residence permit.
Ask about whether you get a multiple exit permit or whether you have to apply EVERY time you want to leave the country, and how long that will take.
Ask about the contract terms for you to resign or for them to dismiss you.
Ask about end of service gratuity and whether they withold a certain amount at the end of your final contract as a guarantee, and if so, when you will get those monies returned.
Ask about health insurance (hopefully they provide private insurance that will cover you both in Qatar and while you're abroad on vacation/at home).
Ask about schooling fees that they cover or requirements for you to send any dependent children to a particular school.

Since you're an educator, I think it's useful to ask about the make-up of the student body, the curriculum of the school, the staff retention rate, pay raises/moving up salary scale (what are the requirements?), professional development allowances/opportunities, personal days, dress code, integration of technology into the classroom/school, language issues (level of ESL), if you can email/contact current AND former teachers there for their feedback...

And I would definitely check out the schools rep on public forums (Qatar Living, International Schools Review, etc.). It's always informative to hear what people think about a school. Expect a fair amount of ax grinding, but you want to hear if it's over-the-top moaning and groaning or whether it's fair criticism.