How can I study in the U.S. with a pending I-485 case?

Until you are approved for your I-485 and lawful permanent residency status, you would still be considered an international student.

Unless your I-485 is approved, you are classified as a non-immigrant. As a result, you will only be given an F-1 status. Further, if you seek an F-1 classification by starting school full-time, you will be treated as an international student and certainly be required to pay fees accordingly until your I-485 is approved.

The rules on in-state tuition vary based on the institution. Some institutions will consider a person for in-state resident tuition if the petition has been pending 366 days in the state, but there are several other factors that need to be considered.

You are not likely to have your permanent residency by May, but check with the school and see what rates they will charge you as an H-1B attending school and/or a PR attending school. Each school has their own policies.

That probably depends on how the school handles international students. A pending I-485, in the eyes of immigration, is not lawful permanent resident status until the final approval, but schools may handle their own tuition requirements differently.