Answers:

1. Yes you can leave company anytime provided you are not bound in any contracts. You can always change job while you are in good status. I did change job when I was in OPT.

2. Sponsering H1B can begin anytime. It depends on your employer (i.e. your consultancies). You have to push them to start the paper work. Once you are in H1B, you will be in comfortable position. It has nothing to do with the clients of your company.

3. It has to be fulltime to be sponsered for H1B. Once you are in H1B, it will be for three years. You can always transfer H1B to new company which is not hard because it is not in quata. Remember- you have to have new paystub if you happen to transfer H1B to any company.