couldn't this also mean "the child would have an animal?" if so, how does one diferentiate? context?
No, that would be "barnet vil gerne have et dyr". "Vil gerne have" roughly translates to "wants" or "would like to have".
So I noticed that the other verbs form the past with one word (spiste, talte,etc). Can "have" do that or does it always need ville?
I think here it's more confusion about the main verb. Think of it more like "The child wanted to have an animal" and it might make a bit more sense. "Have" does have (an irregular) simple past tense of "havde" but here, it's ville which is in past tense (also irregular, and the same as its infinitive)