Do cats and dogs have any kind of interactions between themselves?

As many others have said already: they absolutely do. They play together, they nuzzle/lick each other, one of the cutest things I saw lately was when I aged one of my cats to elder, the other cat had a birthday cake thought bubble and was meouwing, clearly understanding what's going on.

I have households with both cat and dog, and that doesn't stop them from being good friends and playing together and nuzzling each other.

I'd say your pets just need some time to get used to each other. :)