Does impurity transfer from a person who mixes with and touches dogs to those who live with him in the same house when he touches their belongings or them, and how can one guard against that?

A dog's saliva is impure (najis), but the transfer of impurity from one person to another is not decreed except with certainty or a strong presumption. Doubt and obsessive whispers (waswasa) are not to be relied upon.

Therefore, merely touching dogs does not necessarily mean that impurity transfers to everything a person touches. Their hand might have been dry when touching objects, or they might have removed the impurity's trace by washing, or the dog's saliva might not have even touched their hand in the first place. The default state of things is purity, and impurity is not decreed except with demonstrable proof.