請教:小白兔,申請H1B and 綠卡

I just got the job from a university (as a contract librarian),and the university will sponsor my H1B visa for three years.

My questions are
1. I have a common known English name (not official), can I put it in "other used name" field in H1B application?

2. Since my job is contract (renewed year by year), can I ask my employer sponsor my Green Card? or for green card, should I have a "permenant" job? If I could, how long should I wait to ask them to apply for me? after one year?

3. Since I have double master degrees, in my Green Card application, which categery should I belong to?

Thanks a lot!