Bucky's short for Buchanan, so, yeah, it's his name.
Yeah, I guess. Its like calling someone named Robert Bob or Walter Wally, or Richard Dick, or Christine Kitty. But its more like a nick name sounding name than a James. But I still think he can't exactly be called Bucky now, it doesn't really seem to fit him anymore. He has grown out of it. Like, sort of, when Nightwing grew out of being Robin. He didn't alter his costume and become an Adult Robin. So, similarly, he probably shouldn't call himself Bucky anymore.
Did he go by Bucky in the movie? Or was it Barnes most of the time? I don't remember Tommy Lee Jones calling him Bucky.
And I agree, Bucky is his real name, like Nick Fury or Sharon Carter or Jean Grey or Charles Xavier (heh). He maybe could use a code name. That might help solidify his status in the Marvel Universe more.
My theory, even though I'd rather Marvel did something different, is that Marvel will have him do this covert stuff which will take up a year or two. It will move along and take some turns that weren't expected in the beginning and eventually lead to a different place. The different place will probably lead to a different story, which will take up another year or two. And all this time he will probably be called Barnes and Winter Soldier and hardly ever Bucky (by maybe Steve Rogers) and never Captain America or a different superhero name. His costume will probably be his Winter Soldier gear.
So probably he will be the Winter Soldier for another few years and that will sort of be his go to costume instead of being Bucky.
Winter Soldier already has his own action figure.
To me though it felt like Bucky died, but then he returned as the Winter Soldier (who was a bad guy) so I was waiting for him to become good again and become a new hero. But then Civil War happened (and he was trying to come to grips with his Winter Soldier past) and then Cap's death (and he had to take over so no one else would). So, to me, it felt like his character was in limbo for a while trying to decide who he was and what to do, and I was waiting for him to choose something more permanent. I knew he wouldn't be Cap forever, but I thought he might wear that Cap costume without the mask or shield. Just something more permanent to show that he is staying I guess.