I have all the parts of the spaceship built, but I can't seem to have them assembled. I moved a part (any part) to the capital city, but I don't see any option/button to start the assembly. That means I can't move another part in either. Is it because I've already attained displomatic victory?

Yes. If you've already achieved another victory, you can't build the spaceship. I had this same problem on my game, where I destroyed all the civilizations except Napoleon while working on my Science victory, then he attacked all the City-States and they eventually wiped him out.

OK, that confirms it. It makes no sense, though. I still can go on and wipe out all the opponents in domination even with other victory conditions met, but can't finish building the spaceship. In all previous civs it's possible to achieve "multiple victories" even though it's not recorded as such in the hall of fame -- the spaceship would still launch and reach alpha centauri, even if you are the last remaining civ on earth. I'm not whining about "it's not like civ-4-3-2-1, wahhhh!!!11" I take civ-5 as it is; I just think that the designers had actually put thoughts into changing this aspect of the game deliberately, but i can't see the rationality.

Yeah, I have no idea WHY it is that way, I can only guess that the spaceship stuff is directly tied to the victory progress tracker or something.