Well, important things come first. So, that's the negatıon. And, it is negating the verb, so that comes next. Finally, whatever is left.
Kind of an "order of importance".
But the negation does NOT necessarily mess up the word order. It just grabs whatever it is negating, and pulls it in close behind itself. Wherever it is. Sometimes that thing was already in the front, in that case you just attach a negation at the front, nothing else happens.
But there is so much room for flexibility...
I find it a rather basic and very useful word, it can stand on its own as a sentence ("Sajnos."), which makes it easy to express sympathy, empathy and regret. E.g. "I have to go." "It keeps raining this week." "Our flight is cancelled." "You lost the key of my flat." You san answer all these sentences with this single word as a sentence: "Sajnos."
I don't know what your native language is but I think, in English, we tend not to say it very often because it's a complicated word. It doesn't feel like essential vocabulary. I can imagine this goes double for Spanish with desafortunadamente. I was a little bit surprised when I moved to Germany and heard children saying "leider" quite a lot and eventually I realised it's probably just because it's a simple word and we don't use it so much in English because it's bigger and clunkier.
It is actually a very basic concept ... it basically just means you don't like the content of the rest of the sentence.
In English: "I am sorry" is an apology, "unfortunately" is an adverb, so it's changing the meaning of the Hungarian a bit. You could say "I'm sorry to say" but it would be a bit old fashioned, and I would guess there's a separate Hungarian equivalent.
"I am not very good" sounds like you're saying you're not competent at something, rather than that you're not feeling well which is what I think the Hungarian sentence is implying.
Both of these are a bit fuzzy in English, especially the second one - people do answer "how are you?" by saying "Not good, I'm afraid" or something similar. But it's colloquial usage. Technically, jo = good (adjective) and jol = well (adverb), so "How are you (feeling)?" needs to be answered with "I am not (feeling) well".