Yes, you can marry a U.S. citizen and file I-130 and I-485 while on H1B.

H1B is a dual intent visa, so AOS is allowed without issues.

Once you receive your EAD, you can switch employers or stop using H1B.

Yes, you can change from H1B to a marriage-based green card if you marry a U.S. citizen and meet all eligibility and documentation requirements.