1. One week maximum, according to the company you will work with.
2. No.
3. & 4. Once you will get a resident permit, ID or working visa, you can register at Hamad Hospital for only 100 QR, actually your company should do that, but if you want to fasten the things, you can anytime go to a private hospital or clinic, and the registration fee is maximum 250 qatari riyals, including first consultation.
Hope i was helpfull.