"They do not have plates" would translate to "De har INTE tallrikar" which is not quite the same thing as "they do not have any plates" - "De har INGA tallrikar".
How exactly is "inga" different than 'inte"? Is it a word that is conjugated differently depending on the context or is it simply just a different word that looks similar?
It's a different word that looks similar. "Inte" means "not" (e.g. "Jag vet inte" "I do not know") and I think "ingen/inget/inga" translate into something along the lines of "no/none". I'm not 100% sure though, so don't quote me on it!
I read somewhere that inte is used for verbs and Ingen/inget/inga are used for nouns.
I think it's mostly the same as in German. Verbs: nicht, nouns: kein=nicht ein. The point is the included article. Ingen(inte en) and inget(inte ett) have one. Articles with verbs don't make any sense.
Generally I just try to think 'if it's dealing with the amount of something, inga - if it's dealing with the lack of doing something, inte, and so far it has seemed to work :D (I also think of a gold ingot and remember 'that's a THING) and it reminds me till I've had enough practice it becomes natural.
From what I understand inga is used for negating plural nouns, as in here for tallrikar (plates). Inget and ingen are used to negate singular nouns that start with ett and en respectively. All three inga,inget ,ingen usually means "no" or "none" while inte means "not" .