If you are in the US (and this most likely applies to most countries outside the US), you already have a copyright on anything you create. But yes, you can drive the point home by putting a copyright notice on the sheet music which isn't strictly necessary but everyone does it so you might as well. I do this as does everyone I know. It's easy and legal though it doesn't provide any extra protection (except maybe to discourage someone from thinking you've released the work to the public domain).
If you ever need to sue someone for actual money, then you need to register your sheet music with the US Copyright office. If you are planning on making money from your sheet music, this is probably a good idea.
I do not work with ASCAP or any other performance rights organizations. My music is is more in the avant-garde style of classical so I'm not really concerned with performances plus I license my music with a very liberal Creative Commons license so no one would have to pay royalties anyway. All that to say, I don't really have any advice for that side of things.
Now the party begins!
I'm having a hard time finding his post in /r/classicalmusic about it where he and I got into it a bit. It was frustrating, of course.
But most importantly, he holds classical music in such contempt while at the same time trying to appear that not only is he knowledgeable about classical music but that he likes it and is sympathetic toward it. None of that is true. As you note, many of his off-hand comments betray his ignorance as well.
There was a weird phenomenon a number of years ago where it seemed like computer science types were positioning themselves as the top experts in every single field of study because they could dash off a Python script in an afternoon to solve the most difficult problems in all these fields. It doesn't seem as bad anymore but back in the day, whew!, it was a sight to behold! Anyway, Neely seems to have a bit of that about him, classical music can be best explained in terms of jazz and where it can't, it's not worth thinking about anyway.
Yeah, it's just some basic research or maybe even paying attention in class. Oops, he went to Berklee so who knows what he "learned" there!
Adam is always full of surprises!