A tricky question! In daily conversation, we usually say There's a cat and a dog, which is short for "There is a dog and there is a cat". However, it would also be correct to say "there are a cat and a dog", since technically, there are two animals :). But, you won't go wrong if you use the first sentence!

Number 3 and 4 are incorrect to begin with. There're is not a word in the English language. It would be "There are" which cannot be shortened. "Are" is used for more than 1, example, "There are 2 cats and 2 dogs". Number 1 is the best to use as there will be no confusion about how many cats and dogs there are.