"I have no cats" is correct and understandable. Though, I would say, "I do not have a cat." I think "I have no..." is a bit more dramatic. It emphasizes the "no" and makes it feel tragic or sad, whereas "I do not have" is more neutral or factual. I would use "I have not..." to talk about an action; for example, "I have not seen that movie."

"I have no cats" is the same as "I don't have any cats", except it has stronger emphasis. Normally you'd just say the latter, but it's not uncommon to hear the former. You can interpret "no" to mean "zero", and can use this form in other ways (ex. "No cats are..." is the same as "All cats are not...") When indicating possession, 'have' isn't a modal verb like it is when it's part of a perfect tense. "I have gone there" or "Have you done that" are perfectly correctly ( "I didn't have gone" and "Did you have done" are completely incorrect), but "I have not cats" sounds extremely awkward. It was correct in Shakespearean English (when all verbs could be treated as modal verbs), but now it's only used in poetic language. So "don't have" is used for possession and "have not" is used with perfect tenses.

Yes, you can say that, but you have to realise that 'have no' is not a verb construction. The verb in this sentence is simply 'have'. The word 'no' is part of the object - 'no cats'. So the two sentences (I have) (no cats) (I don't have) (any cats) have the same meaning, but slightly different grammar. In the first it is the object which is negative and in the second it is the verb.