Thank you novita77 for your useful info. In my country (and Europe too) the public health assistance is free of charge. You pay only a low fixed price (ticket) for every kind of assistance.
The National Health Care - to which belong hospitals, out patient departments etc. - guarantees medical assistance to all citizens, Italian and foreigner without discriminatory treatment. In order to benefit it is necessary to enrol, and the sanitary card (with the name of the beneficiary and of the doctor in charge) proves the application for admission.

Sanitary card gives the right to:
1. have a family doctor and paediatrician;
2. have specialist assistance, at the cost of the ticket;
3. hospitalization either private (if operating within the national Health Service) or public;
4. assistance for buying medicines;
The medical assistance is guaranteed to everyone who is enrolled and to their families if they have a regular residence permit.

I imagine that I need a private health insurance for me and my family because I do not think to be a resident in Q, but to stay a couple or 4 years only.

You say 10.000 QR for your son, I hope it was for the childbirth and not for an yearly health care fee!

Which kind of school have you considered for the price of 6.000 - 7.000 QR per month? It's an high fee: 20% of the salary.