I self-published for a number of years, so I can’t say from a traditional viewpoint. But, so far as I understand the copyright isn’t an issue, so long as you can prove you’re the original author (usually meaning that you have the original file/s).
As far as explaining it to a publisher, plenty of people use pseudonyms. Half (at least) of the romance genre is written by men using female pseudos, and I imagine quite a few crime novel are the opposite. Also, sometimes people just don’t want to have their name out there, like you said. Whether it’s more because they are afraid of people trying to use their name, or personal preference, I can’t say. For me (my name on here is a pseudonym), it’s a personal preference thing.
And no, you don’t have to copyright the pseudonym. Actually, titles (and in this case, names) can’t be copyrighted. That’s why some books have the same title.
As far as it being realistic, well...I think it used to be more of a benefit, but now I’m not sure how much it matters. It also depends on what you’re writing. Each genre has its conventions when it comes to names, both for the characters and the author. So, if you write fantasy, a slightly out there name might not sound weird (or, you may be inclined to use initials). If you have a following online already, before you come to the publisher, that means you’re marketable, so I don’t think they would care. I’d choose something you like. Something that is, like the above said, easy to remember and marketable (and that you hopefully won’t hate when you get older).
Hope that helps!