Can the word paingry be used only as adjective?
"I was too paingry to react"
Assuming we can compare Paingry with similar words such as angry and hungry, then these are nouns with the suffix -y added to make adjectives (anger + -y => angry, hunger + -y => hungry etc). Wiktionary describes this as "Added to nouns and adjectives to form adjectives meaning "having the quality of"".
So from that perspective, one would not normally use a word like paingry as a noun and by analogy with angry and hungry you might resort to "painger".
That said, from a descriptivist perspective I think this word is too rare to have a "standard usage" and any usage is pretty valid. If you want to start using it as a noun and you think people will understand what you mean by it, then it can be used as a noun! But it's probably wise not to use it in an exam or very formal setting as there's a good chance people won't understand your novel usage of a very unusual word and you don't want to blow a job application or anything :-)
I just thought of a standard way to use it as a noun! By placing it in a clause like "The paingry", it becomes a noun phrase referring (usually) to people who are paingry. For example:
I get on well with the hungry, but terribly with the paingry.
I would definitely caution any English language learner who wishes to use this word that it's a very recent construction, and also one that isn't really obvious on its own — it relies on the listener being aware of the other recently-popular "joke word" hangry.
"Hangry" is a portmanteau of "hungry" and "angry". These are both adjectives with corresponding nouns (and verbs, for that matter) "hunger" and "anger". But "paingry" appears to be a combination of "in pain" and "angry", which is less clever/elegant because the words don't share an ending.* The Urban Dictionary link you give suggests that this is meant to parallel "hangry" anyway. Urban Dictionary isn't really a dictionary, though, in any traditional sense — it's a website where people can make up whatever they want.
A perfectly reasonable alternate definition from just looking at the word alone might be that you mean "painger" to be a new word parallel to hunger and anger, meaning "a feeling of being in pain". Or, maybe it could be meant to be a combination of hungry and in pain. There's no real way to know — the primary reason one would assume (as I immediately did, for what it's worth) "pain + anger" as the meaning (without reading your Urban Dictionary link) would be familiarity with hangry.
So, were you to use this word, I would generally expect you use it along with a following explanation:
“Sorry I yelled at you when I stubbed my toe. I know it wasn't your fault. I was just paingry — you know, in pain and angry."
This was definitely common when "hangry" was younger, but it seems like that one is settling in to being an informal but not uncommon expression. I don't know if that'll be the case with "paingry", which seems like a less useful word overall.
Because hungry and angry are both adjectives, it follows that "hangry" follows the same rules, and that "paingry" does too. That means that these words can function as nouns.
* Tangent alert! "Hangry" is specifically interesting because there's a clear connection to an old, insoluble riddle which claims that there are three common words which end in "-gry" — when in fact there's really just the these two. But now, perhaps, there are three, including hangry... and then, in the unlikely event this catches on, four!
This is a made-up word, but it's presumably derived from pain + anger. So I'd guess the noun form would be "painger" and the adver form would be "paingrily".
I would guess that this word will not catch on. It doesn't express any new idea, just the combination of two existing ideas. It's pretty much useless. I hadn't heard it before reading this post, and I won't be using it. The words that tend to "make it" are the ones for new ideas, where it would take a whole paragraph to explain what you were talking about without the new word. Like "e-mail", "boson", etc. We'll see, I suppose.