H1B transfer from university

Dear Rosalinda and others who want to help:

I just get an offer from a private company. I am working for an university and holding H1B visa. The company has agreed to apply H1B for me. The following are my questions:
(1) They said it costs $4000 in total, the company pay half, I will pay hal.Is it true? I heard that H1B only needs $1000 application fee(for premium processing)

(2) What is the formal procedure to apply a new H1B, what should I do? What kind of documentation INS need from university's international office?

Thanks and have a nice evening