Yes, ludo can be used to mean to play music. The line I cited from Vergil says "... ipsum ludere quae vellem calamo permisit agresti", which roughly means "and he permits me to play what I like on my country reed-pipe/flute".
I'm not sure about facio; although that is not to say that it could not be used. It has many meanings depending on the context, but, very generally speaking, it has more to do with manufacture, creation, production or composition of something. I know the phrase 'carmina facere' means to compose or write songs/poetry, rather than singing or reading out loud.
Regarding the classical violin makers, i think 'faciebat' refers to the making of the violin, not playing music on it.
The noun is musica. Musico is ablative or dative of musicus, an adjective.
“Cum ego verbo utar,” Humpty Dumpty dixit voce contempta, “indicat illud quod optem – nec plus nec minus.”
“Est tamen rogatio” dixit Alice, “an efficere verba tot res indicare possis.”
“Rogatio est, “Humpty Dumpty responsit, “quae fiat magister – id cunctum est.”