"Nyitva" is definitely not an adjective. I think that is why it is not called an adjective. :)
And yes, it is "határozói igenév" (adverb-something-verb-something), because it takes on some characteristics of both adverbs and verbs.
Here is some extended discussion on adverbs and their relationship with "van".
I have seen that discussion a while ago, but now I understand a lot more. It's fascinating.
So, -va forms are not adjectives. To be clear, that means you cannot say things like "a nyitva üzlet" but would have to resort to using nyitott, right?
Someday I will crack the határozói. Just needs a lot of practice. (v °o° )^
That is exactly right, you got it!
Other ways you could say it are:
"Az üzlet, amely nyitva van"
"A nyitva levő üzlet"
I guess "levő" (or "lévő") would be what you call the present participle of "lenni" ("to be").
"Nyitva levő" - "open-being", literally.
So, not an adjective at all.