Hi there, great question. And a very important one in the world today!
Firstly, yes, almost every animal carries and spreads diseases. However, these may only be harmful to certain other animals. Take for example the common cold. We humans are affected terribly by this virus, but birds are not! This is because they are not very closely related to us.
Where I work, gorillas are at great threat because they are so closely related to us, meaning that the sicknesses that we get will affect them badly and may kill them. So when people visit gorillas they must stay well back and if you have a cold or flu you cannot even go! The reason that mice and rats spread so many human diseases is because they have lived with us (well, by eating our food!) for millions of years. This means that any diseases they do have will affect us more as they’ve evolved to attack humans! But I wouldn’t worry about it, as medicine these days can handle most of what rats and mice can throw at us!
Cats and dogs do spread disease yes! Cats can spread toxoplasma and dogs (well, their poo…..) can give you toxocara which damages your eyes. Again because they live with us, these diseases may be getting worse.