"Families" is the plural of family, that is more than one family. Your sentence is correct. "my family" refers to all of the members of your family. There are refinements "immediate family" usually means mother, father, and children and "extended family" all other relatives i.e aunts, uncles, grand parents, etc.

"family" already includes all of your family members. The plural would mean two (or more) different families. Your example sentence is perfect.

No, you can't. 'Families' means two or more separate families - for example, "The students all went home to their families for the holidays". 'Family' refers to a group of people, not individuals, just like 'team' or 'committee' or 'staff'. The individual players, for example, are not in themselves 'teams' - they are 'team members'. This error is common among Chinese learners. Is this because of first language interference, I wonder? It's fine to say " with my family ".