You don't get your contract to sign until you actually arrive and pass all the health checks.  The letter you have should specify accomodation provided (or allowance), tickets, health, salary, work terms and conditions - and reputable companies will provide a contract (after the health check) that match the letter. 

The contract isn't offered at this time because if you don't pass security and health checks you won't get a work visa and will have to leave Qatar.

If you think you're going to be interacting socially on a regular basis with Qataris, you're mistaken - most Qatari families keep to themselves and their social group, however, if you are in charge of a nursery with Qatari children in attendance, you may have more opportunities to do so than most expats do.

If you feel the need to negotiate, do it now, you won't be able to do so when the contract is presented after you arrive.