Agreed - I missed that.To answer the simplest question first, the "of" is optional. The sentence needs an "a" or a "the" before band, though.
I don't understand what you mean. Things that are literal can be idiomatic If things are common they are almost certainly idomatic (again confused).Second, the expression "very musical family" seems pretty literal to me, so I would say it is not idiomatic. It is rather common.
I don't understand what you mean - I suspect you are assuming that idiom equals "set-phrase". If so, it doesn't necessarily - your link doesn't work so I can't check your misapprehension, but idiomatic usually means "an acceptable marriage of certain words in a phrase". You are mixing up meaning 2 and 3 here . In any case, "she comes from a very musical family" is a perfectly idiomatic phrase to my understanding of the word. Let's see what others think.
To answer your edit - Why would you necessarily use the second one in formal or academic writing if both (quoting you) "will work fine and have the same meaning as a pronoun or an adjective"?The only situations where you need the "of" is when Both is followed by another pronoun.
Both of them are studying math.
Both of them are in the choir.
In most other cases, both will work fine and have the same meaning as a pronoun or an adjective.
Both of the girls are fine.
Both the girls are fine.
I would necessarily use the second one in formal or academic writing.
--Is "a very musical family" very natural and idiomatic in the following sample?
1) Yes, it is pretty literal, and it is rather common.Second, the expression "very musical family" seems pretty literal to me, so I would say it is not idiomatic. It is rather common.