Hungarian Residence Permits

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I'm a US citizen living in America, My girlfriend is a Hungarian citizen living in Hungary. We wish to live together in hungary, for at minimum the next year and a half, as she's in a Master's program at university and needs to stay in Budapest to complete it.

We're not yet engaged or married, however these things are not out of the question either. Largely because it seems equally daunting to figure out the best way to make such a thing happen.

What are our options for long term residency, and the easiest path to achieve it?

In america i'm unemployed and on disability. Which should stop i presume after the move, however if necessary I can have a substantial bank account as proof of sustainability.

My primary questions are, giving my as of now inability to work, what 'proof of purpose' could i present in an effort to stay? is the only way to marry and request it under family reunification?

Also, how do i obtain Hungarian medical insurance? Here i have both Medicare and Blue Cross Blue Shield, but will I have to somehow obtain and pay for new insurance in Hungary?

Also when presenting my bank assets as proof of sustainability, does it have to be a hungarian bank?

If this breaks board etiquette somehow, the nuances of your community, i apologize; i've just found this site. I haven't been able to get questions answered by embassies, and just want to figure this out.

Lastly, is it correct that under the 90 stay visa, you can visit up to 90 days over multiple periods within a 180 day window, after which this resets? So theoretically i could say 1 month, leave 1 month, stay 2 months. Wait for 180 days from first visit to expire and then repeat?

I realize these are obvious questions to the informed. But my brain reads it as legalize and confuscation, and i'm worried about the delicacies and hidden words in a multipage lawyer text.

Thank you.