...who exactly is the Assassin, who is he working for, and what is his goal? ...is he [Regina's] love interest?
where is [the Assassin] from, what time period, what is his occupation, does he have a love interest, etc.?
Well, that's certainly all-encompassing! For a general introduction, the best place to go is his Beautiful People post; I wrote it five or six months ago, but reading over it now, I think it's still pretty accurate. Not that he hasn't developed and changed since that point in the story itself, but his personality - likes and dislikes, looks and habits - remains the same. The very old excerpt, on the other hand, has been completely overhauled and no longer provides an accurate picture of either the Assassin or Regina. I'm considering posting the revised version (though I'm sure it will change again in the drafts to come).
So then, who is the Assassin? He's a big, cheerful, often childish fellow who has traveled the Western and Eastern Empires and somehow ended up in Britain, possibly because it was the only place in the known world he hadn't visited before. Throughout the story he is known by a variety of names, including "the blond man," "the man like Atlas," and, yes, his real name. He has a generally sunny outlook on life, in contrast with Regina's pessimism; and aside from his hazy background, there is nothing about him that would bring his occupation to one's mind. His path crosses with Regina's by chance, as far as she knows, and he promises to serve her and get her back to her own time - in exchange for a promise that, once he does, she will give him the pocket watch. On the surface that is his goal, but what he really wants is always a variable.
At the time when "Regina's personal continuity intersects with his personal continuity," the Assassin does not appear to be working for anyone. If he had been, however, his employer would have found himself dumped in a heartbeat, for the Assassin is first and foremost...something other than an assassin. Killing or quack-salving pays the bills, as it were, but they are not the love of his life. If he suddenly inherited a fortune from all those relatives he doesn't have, he would probably give up at least one of the two vocations. I'll leave it to you to decide which one.
Love interests! I see you are trying to sneak in the back door and eliminate the possibilities one by one. You will next be asking whether the Time King has any love interests, or whether Regina's could possibly be the White Demon, or whether the Fisherman...! They are all, I think, better bets, for the Assassin is not your typical love interest material. He is frequently so childish, so lost in his own happy world of diagrams and theories, that whether or not he recognizes Regina as a woman is a debatable point. Remember that he was partially inspired by Sherlock Holmes, and then don't get your hopes up too high.
Does the Assassin regret any of his kills?
(To avoid spoilerisms, you can limit this to kills before the story starts.) (But because time-travel, before is problematic, so, to kills before the intersection of his own personal continuity with the personal continuity of Regina, or, in the event, the personal continuities of any other important characters with which his own personal continuity has intersected or will intersect in any sort of way, timey-wimey or otherwise.)
But don't worry, he's comforted in the knowledge that once he does, his future self won't regret it.