As a single sentence, "Те дали имат нещо против?" is a question ("Do they mind?") and it can't be used without a question mark. Translating it as a statement ("They do not mind.") is definitely wrong. It can be used as a subclause in a composite sentence though: "(Чудя се) дали имат нещо против." "(I wonder) if/whether they mind." Since such subclause also is a kind of question, there's no big difference whether it's "mind" ("имат нещо против") or "not mind" ("нямат нищо против")

Hello Paul! This sentence literally means: Те дали имат нещо против? They if have something against? Since in the sentence there is the word "дали" this means that the sentence is a question. I cannot think of any situation when this sentence will end with a full stop. And to your second question, once you have used "дали" in a sentence you do not have to use "ли" anymore. Of course you can use different word order and ask "Те имат ли нещо против?' or "Дали те имат нещо против?". There is of course no change in the meaning!