Q. I am a student graduating in December with my Masters degree. I have an offer from an university. I understand that universities apply for exempt-H1B. I also understand that exempt H1B's cannot be transferred to non-exempt H1B's.
Will I be able to work in the period from April to October 1st with the private company, when my non-exempt H1B is still in process? Will accepting this offer be a problem if i want to move to a private firm later?
Ans. The regulations seem not to address this situation. If I were to decide purely based upon the language of the regs, I would guess that you can work.
But, my GUESS also is, somewhere along the line, CIS will clarify that you cannot work under these circumstances. The new regs were put into place to eliminate gap in the employment that occurs when a student and their employer confront the gap in employment between expiration of the F-1 EAD and October 1st - start date of the H-1. This may not be applicable to universities who are quota exempt. In your case, you will be working for a private employer while the H-1 is pending through the university. Tough call as far as I can see.
Do note, I have not spent a lot of time thinking this issue through. So you should ask your employer's lawyers to give an opinion in writing.
Q. Also will it be possible for me to request the university to apply for a non-exempt visa for me? Is that an option?
Ans. This would be an option if you were actually not working for the university, but with an unaffiliated entity. I do not see how the university can apply for a non-exempt H-1.