'The 'of' a Friday' doesn't make any sense to me, I would suspect it's either a typo, or if it was spoken verbally, it's a person speaking very quickly and and annunciating. It would either be '...go out on Fridays' or '....go out on a Friday.'

OF A FRIDAY, is correct, but pretty rare and possibly old fashioned in most of the US at least. OF A FRIDAY would mean something like SOMETIMES ON FRIDAYS. It's good to understand this, but I wouldn't advise using it yourself - as you see in the responses, even many native speakers are confused by it. PREFER DOING and PREFER TO DO are essentially interchangeable, in my opinion. Unfortunately, TO DO and DOING are sometimes interchangeable, sometimes not. MUSIC LOVER is a common term, in answer to your last question.

Friends of a Friday sounds unusual to me. Perhaps it’s a dialectical usage. As for your second question, I don’t think there is much if any difference. For example: I prefer to eat alone and I prefer eating alone seem to be the same to me. The third question is a little easier: you can just say that person is a music lover. If you want to sound a little pretentious, you can call such people music cognoscenti (from Italian) to mean they know a lot about music or music connoisseurs (from French) to imply they are very discerning in their musical tastes. But music lover is just fine.

Question #1: The extra syllable "a" can be inserted before just about any word as a space filler or emphasizer. It is used, but only occasionally, for color in casual speech: "I love to go a-wandering along the mountain path" "It ain't the leavin' that's a-grievin' me / But my true love who's bound to stay behind" In your sentence, "of a Friday" means "on occasional Fridays". It is less precise than "on Fridays". Question #2: The difference between present participles ("doing") and infinitives ("to do") is that participles feel more descriptive and infinitives feel more factual. The meanings are exactly the same, but the feeling is different. If you want to paint a picture, use the participle. If you want to get the facts across, use the infinitive. For example, compare "I prefer to do the cha-cha. I don't like the samba as much." (These are just facts.) "I love doing the tango because the rhythm feels so good and it fills me with romance." (This is an image. It makes you see the person dancing.) Question #3: A music aficionado (refined) A music lover (standard) A music freak (colloquial) A music addict (not good though to be called an "addict") A music devotee (nice if you like sounding French)

It is very uncommon and I am not completely sure, but I would guess that "friends of a Friday" means friends that see each other on Fridays. I suspect it is either an archaic expression or poetic. There is an expression "birds of a feather, flock together" which means people with similar interests tend to stick together. That is the only time I have seen this sort of "of a" construction that I can think of. I prefer to see movies on the big screen and I prefer seeing movies on the big screen are identical in meaning. If there is a difference, I can't think of it. A musician listens to music all the time and is fond of it, but also makes or composes it. Someone who just listens but does not play an instrument or compose would usually be called a music-lover. Musicphile is a word apparently, but I have never seen or used it until today. An audiophile is someone who loves stereos which of course are used to play music, but they may be more interested in electronics than the music itself. Music afficionado is also possible but music lover is by far the most common expression. I hope that helps.