I'll only give this a brief glance now as I'm very tired, but I'll put my head on the block and say all white pieces fail to work (that's simple to work out), and if the black bishop works, that is one fine piece of pawn geometry you got there. I actually see no concrete reason it shouldn't be legal with bBh2, but it's all dependent on a piece of good timing that I can't judge without trying.
Is this position legal?

random thought, but replacing the Nh2 with the black dsb should make it legal, the capture colors should work out now. You put both bishops in the corner, push g2, send the king in, promote pawn (white knight in g2?)

#428, yes. black has a knight (or anything else really) hop around while the queens get set up, then the last move would be one of the queens taking the knight on the square.
In retrospect, having black's remaining piece be a rook and move from b8 to c8 looks a lot better.

Ah, the classic colour of capture squares. I second mashanator's analysis (and unfortunately, sentiment. Retroanalysis does not appear to be in most chessplayers' repertoire of skills.)
@ghostqiuyu #420 trivially legal, after retracting, say, -1...Rh8xQh2 -2. Qb8xBh2 and both sides have as many moves as they need. Impossible-looking checks are not necessarily impossible, at least once you know the capturing and en passant tricks.
It is not as grim as that - there are actually quite a few people here who have retro skills: you, mashanator, chaotic_iak, summersolstice, shoopi, omnipaul, tmkroll, etc. You have to be patient with newcomers and walk them through the common tricks. Most will abandon the genre after discovering how tricky it is, but one or two might stick around and get good at it.
Holy! I've tried to solve it but it's really hard, notso much because of the a1 rook but rather because of getting the king on the right place. I either couldn't get there because of the bisshop or because of the Queen. Even putting a black Knight in the bisshop's place is no option, because you either check the king, which can only be solved by capturing the knight, or via b3, but then you have to capture the bisshop which has to be replaced by a promoted pawn, which is impossible because there are too few pieces so that the h-pawn can promote on a black square. I am sure there is a solution, but I don't know were I went wrong.
My compliments Remellion, it is a great puzzle again.

@Cyklope: #428 illegal with white to move, legal with black to move. And you seem to have retro skills too! The more the merrier.
@chaotic_iak: Now I've had some sleep. Solution PM'ed. Very nicely conceived.
@BigDoggProblem: Yeah, that's true. There are many strong solvers/composers lurking around. Getting the uninitiated familiar with the common tricks is kind of tricky (...) since it needs lots of time and basic positions. I suppose the best way would be to provide simple problems and outright say "this is legal; what happened?" rather than letting them determine legality themselves.
@lixinzheng and Cyklope: Sure. It was actually a twin of #287, which is quite a bit easier and probably should be tried first. I don't think many more will get this one so here's one solution; the trick is to get an "extra" black piece from where it's not needed.

Now I suddenly feel like wanting to start a thread for introducing people to retro problems (by giving really simple stuff)... Not sure how many people will be interested though. Also, not sure a forum thread works good for that.

I've been following this thread for a while, haven't chimed in much because people keep posting the solutions before me (I will never forgive you). There have been a few that I couldn't get, like the 6 pawns in a row for both sides, and some of Remellion's really insane ones. I am by far not the best solver here! I would find a thread with common tricks and motifs pretty nice, especially since I am having problems composing positions. Also, if anyone has tips for composing, I would like to hear them very much

OK, I've started an intro thread for retros. Click here.

442- took a first glance, then realized some interesting things. I'll get back to it.
443- trivially legal
manshator: Correct. And so part b:
b) What color/piece, if any, can replace Nh2 to make the position legal?
(Clearly if it remains a black piece, then nothing but black bishop can replace it by the analysis. It remains to show whether with black bishop the position is legal, and whether using a white piece works...)
Cyklope: Good analysis, but as specified before, Rf1 is promoted from g2xf1 and was a pawn. There are six remaining pieces where axbxcxdxexf7 can take five. It's only a step away.