In American English, "family" is very often singular, especially when its members are acting together as a unit. In British English, "family" is often plural. In American English, your sentence would most likely be "My family is," and in British English, "my family are." Both are considered correct.
Check out this brief Q and A: https://thegrammarexchange.infopop.cc/eve/forums/a/tpc/f...986031765#9986031765
And this one: https://thegrammarexchange.infopop.cc/eve/forums/a/tpc/f...171039833#6171039833
As for "house," it belongs in the sentence. Don't leave out "house." It is the object of "is/are painting" and needs to be in the sentence to complete the thought.
Michael Swan* notes that British English uses both singular and plural verbs with many collective, or group, nouns, such as family, team, committee, government, and class. He also notes that in American English, the singular form is normal with these nouns, although family sometimes takes a plural verb.
*Practical English Usage, by Michael Swan. Third Edition. Oxford University Press. 2005