I did this 10 years ago, so my answers may be out of date.
- should a non-EU national (who's a spouse of a EU citizen) apply for visa/permanent residence card in their own country (through consulate) or at a local immigration office in the host country?
This is really something that should be asked at the Hungarian consulate. As I understand it, the process should often start in the home country of the non-Hungarian spouse.
But there are exceptions: I moved to Hungary from Switzerland, where I lived for many years with my Hungarian wife. In Hungary I was told I needed to go back to the USA and apply from there. I wrote letter saying due to having lived in Switzerland, I had no place to return to in the USA and asked for release of this requirement, and it was granted, so I was allowed to apply for residence in Hungary.
- how much money would be considered sufficient on their bank account?
I do not know. But I think they just want to make sure the applicant will not become a social case in Hungary.
I was asked to submit two sequential bank statements. I think they were more interested if I could show a regular income to prove I would not become a social case. I kept accounts in three different currencies, but just supplied the one which had regular deposits, not the one with the most cash. So I guess the amount in the account and the payments were "enough", and both were what one may call "average" for most of western Europe, but ironically below what one would expect in Switzerland .
- do they need to apply for/pay for health insurance (for Hungary) before applying for the visa? Does it need to be 3 or 12 month of insurance?
Can not help much there. There was no health insurance requirement when I applied. I had to get it later when the law changed, but by that time I could register with national health.
- what documents will the applicant and their (EU) spouse need when applying?
There were a lot. The file I eventually amassed was very thick. I can not remember them all, but amongst the paperwork included proof of marriage, passport copies, birth certificate (my current residency permit actually lists on it the city where I was born which they got off my birth certificate), financial documents, proof of no criminal record, university diploma, registered proof of where I would live, just to name a few.
- do the documents need to be translated or just notarized?
All mine had to be translated, and I was required to have it all done by an official Hungarian translation office (and had to buy a bunch of stamps which were affixed to the translations to make the translations official, a notarization was not sufficient).
I guess what I need to know before my husband would make the very long and expensive trip back to Hungary is that whether it'd be better for his application if I traveled home sooner, got a job and applied only after that.
Funny thing is, my wife had to sign a statement that she would be financially responsible for me, even though she was the one who quit her job in Switzerland, and was unemployed and had no income in Hungary, while I still had a regular guaranteed payments from US sources that did not even require me to work. In reality, it was I who mostly supported our post move here in the beginning. On paper, she did. Just an example of non-reality based hoops to jump through.