Do you like animals? Do you have any pets?
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I like animals and no I don't have any pets .
I like animals, but I do not have pets
I like animals, but I haven't got any pets.
haven't got=don't have
I like animals and no I haven't any pets.
I would use "any" (unlike Funnyosa), although I don't think it's wrong to omit it. We often use "any" for plurals when using negatives and questions.
ValTheRock's suggestions, and Polbol's suggestions are as good as if not better than mine.