Grammatically it's fine. I'm not sure I would say 'burn down' with plants, as they typically aren't very tall. I would say burn down trees, or just burn plants.

I think it's fine as written. In the United States we would call them "wildfires," one word. Oxforddictionaries.com is telling me that "bush fire" is spelled as two words, with a space in between, but a quick Google search turns up so many examples of "bushfire" as one word, in newspaper stories, that I believe Oxford Dictionaries may be behind the times. I don't believe there's any fixed rule about using more than one "and" in a sentence. The only rule is that the sentence must be easy to understand. In your example, I don't see any problem, because there is no possibility of perceiving "houses and plants and kill" as a list of three nouns. I think I would put a comma after the word "plants," but I'm old-fashioned and tend to use more punctuation than other people.

And should really only be used in a sentence once. I would break this in 2 sentences. Bushfires may burn down houses and plants; They also kill people and animals. As the sentences are closely related a ; can be used showing that the sentences regard each other.