How does paying after insurance coverage really work?

Background context: I’m (22F) not living with parents but am still under my father’s family plan (Oscar) in Florida, United States. Never grew up with information given to me how insurance works, had to request multiple times to get my insurance card out of my father.

Recently I had a corneal abrasion and was at the ER on a Sunday. They asked for my insurance and they let me walk out without paying anything. I’m extremely skeptical of the visit being completely covered. My boyfriend mentioned to me in a passing comment that they most likely charged my father on his account and I have no idea if this is the case or not.

I also use Oscar for my birth control medication on Twenty Eight Health and I pay about $12/month on my own.

My boyfriend is on his own insurance thanks to his job. I noticed that his insurance covers a partial amount and he pays the rest. This is the way I thought insurance normally worked.

I’m asking here because I really need to step up and start taking care of my health because I’m extremely behind/unknowledgeable about my records thanks to my father. I don’t even know what blood type I am. I’d like to get bloodwork, dental checkup, PAP smear, and overall regular healthcare done.