H1B transfer from university

H1B transfer from university to company

One contract comapny offered me a position, and said they will do the H1B transfer.

(1)However, current my H1b is sponsored by a university and I never worked in a company using H1b, can this h1b still be transferred from University to company?
(2)In addition, they said that I can begin the position right after we receive the receipts of H1 b transfer from USCIS, is this right?
(3)If H1b can not be tranferred, what option do I have?

Thanks