The first one is correct. The second one should be: “ my family HAS horses.”

The correct thing is to say : My family has a horse OR My family has horses.

#1 is correct if your family only has one horse. I think it is a tendency of British English to treat a group, in the singular, as a plural: "My family have..." "The council decide..." etc. It depends how you're thinking of it. As an American, I would treat the group as a singular, because I'm thinking about it as a single entity. You'll have to check British English standards, I'm just going from something I saw in a movie a long time ago. So if my family had more than one horse, I would personally say "My family has horses."

The statement 1 is true. The verb depends on the subject.