openings for their kids at even a mediocre school is often quite difficult here in Doha.

Re: the kids attitudes at the schools not reflecting a truly Islamic kind of perspective -- you're going to find that most places, I think. At least with the better schools (that I mentioned above), they have their kids do community service projects and the schools maintain a decent level of behavioral expectations that you won't find elsewhere.

I think finding a qualified Islamic studies tutor is simple enough in Doha. I would use word of mouth or head to the Islamic Center (Fanar) here for recommendations, but I suspect that will be easier than finding a good school (as you've surmised!).

If your kid is 5 that equates to Kindergarten in the US system, Year One in the UK. I stand by the earlier schools I recommended as being the best academic ones that I would try for.
Most probably have waiting lists -- but if they give you the sense that there might be an opening in a few months, etc. I think I would opt to play the waiting game, enroll them somewhere temporarily for now, and then make the jump when you can. When your foot is in the door, you've got a secured place (and an advantage for any future sibling placements usually) unless your kid struggles academically/socially and is asked to leave (which hopefully wouldn't be the case).
And for schools like ASD, getting in at the beginning (when they're 5) is FAR easier than trying to score a spot down the road when all the other kids are simply continuing along, having already secured their space at the school.

By January, you should also probably be filling out applications for NEXT year (2010-2011) at various schools (testing doesn't usually happen until March/April though). Stay in touch with the schools that top your wish list and make personal contacts with the registrars. A lot of these schools ALSO greatly value parents that present a positive image as adults who will be consistently involved in their child's education (attend meetings/concerts, regularly stay in touch with the teachers, etc.). So don't forget that in many cases YOU are auditioning as well!

"If you're looking for sympathy, you'll find it between sh*t and syphilis in the dictionary."
- David Sedaris