Based on my guess:
Question 1: I currently work as a post-doc in an university. I have an EAD based on F-1 OPT and also an EAD based on I-485. Which one should I use for the job?
Question
YOU SHOULD USE 485 EAD, AS YOUR OPT EAD WAS INVALID AFTER YOUR 485 EAD
2: what paper works should I have with this university?
BRING YOUR 485 EAD TO YOUR PAYROLL, THEY WILL LET YOU FILL NECESSARY FORMS
Question 3: Can I apply for H1B visa if I work using the EAD based on I485
YOU CAN, BUT YOU NEED TO CHANGE H-1 VISA OUTSIDE OF USA