"Was" is grammatically correct, since "family" is a singular noun, as Rivka said. But plural verbs are fairly common (especially in Britain, I believe), and it's especially common to refer to "family" with the plural pronoun "they".
Rivka's also right about the question marks. In the first you're questioning a quote, not quoting a question. In the second the whole parenthetical sentence is a question. Generally, punctuation only goes inside quotation marks if it's part of the quotation, but in American practice periods and commas always go inside, whether or not they're part of the quotation.