Qatar Academy (starts with 3 year olds, IB all the way through), American School of Doha (offers IB program for grades 11-12 only, and also runs AP courses), Gulf English School (grades 11-12 only, too)...and four others (but I don't think the other four would be suitable for your child; these three are the best choices for expat kids).

Both Qatar Academy and ASD are regularly full and have waiting lists -- so contact them NOW and check availability/waiting list situations. ASD's high school costs are approx. QR 58,000 (+ extra fees of around QR 18,000). QA's high school tuition ranges from QR 32-40,000. You'd probably find ASD more similar to the schools in the States (it's at least 50% American in student population, whereas Qatar Academy is actually a national school with 50% Qatari students and 50% mixed other nationalities - the majority of whom are expat Arabs; QA also requries uniforms). Academics at both schools are great, I think, but extra-curriculars at stronger at ASD.
Admission at both schools requires previous school records, a personal interview and an admissions test (to demonstrate that their math/English skills are where they should be for their grade). They will not offer you a spot without first seeing your child personally and going through all of this.

Both schools have functioning websites with pertinent information.

I can't speak to Gulf English -- don't know it well enough.

There are many nursery schools around. Choose one once you get here and know where you will be living; you'll want one that is both good and close to home (traffic in Doha is ridiculously bad now).

Sidra is an option but there are many other private medical centers/hospitals here, and as you have already found, most aren't great about having working websites or even about answering their phones (frustrating to say the least). Better to wait until you're here and can investigate what's out there.

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