well my 2 comments got upvoted, so who knows, maybe i can turn this into an answer.
(Note: I wasn't really into chess or chess960 at the time I read the book (except that I played a few tournaments as a kid), so wow this was a little nostalgic. When I did get more into chess/chess960, I read about the silman position. That's about it.)
jeremy silman talked about the position in these 2 links in the description here. unfortunately, they're both dead. let's see if i can dig up web archive. oh yeah here we go
(also http://web.archive.org/web/20170801000000*/jeremysilman.com/shop/pc/Creating-the-Harry-Potter-Chess-Position-87p3692.htm and http://web.archive.org/web/20111201000000*/jeremysilman.com/movies_tv_js/harry_potter.html)
Also for reference here's the position: 5r1k/1pN1R1pp/1Pb5/n1r1P1n1/7N/b2p4/7P/1R1Q2K1 w - - 0 1
aaaand of course there's agadmator: Ron Weasley Sacrifices Himself for the Team
anyway, let's see if the game in the movie satisfies the conditions
White starts with 1. e4
- Yes. Ron does Scandinavian defense.
The white queen takes both black knights
- No, just Ron. But the other knight doesn't really do anything, so I don't see why there couldn't have been a position where Harry, Ron and Hermione have lost their other knight. An unlikely thing is that there was an early underpromotion.
The second black knight (Ron) sacrificed himself to the white queen in order for a bishop (Harry) to travel 3 spaces to the left for mate
- Sort of. Yes, assuming (A) you don't count the intermediate move of capturing the queen (the movie didn't seem to count this!) and (B) if you count moving 2 squares diagonally as 3 total, like there are 3 numbers between 1 and 3 inclusive but it takes only 2 (+1)'s to go from 1 to 3. (referred in (7.2))
The first black knight was taken for a trade for a black rook (Hermione) to take a white bishop
4.1. see (2). Well, white doesn't have any bishops left, and Hermione maybe only moved a little after they presumably castled. Maybe there was a Bxf1 and then Hermione recaptured. However, Hermione has control of this really critical f-f-file to the point of blocking (I guess I don't mean 'blocking' in the chesstempo sense) the exposed white king's escape s.t. Hermione, Harry and the other bishop ultimately checkmate the king together. In fact, it's because of Hermione's control that at the start they already have this mate threat with a combination of Ron, Hermione and the other bishop.
4.2. Btw in the movie, Ron says Hermione is the queenside rook ('queenside castle'), but it seems here Hermione is the kingside rook (unless maybe Ron did some kind of rook lift)
4.3. Hey funny thing I noticed re Hermione and Ron:
4.3.1. After Ron's offered rook (the non-Hermione rook) sac Rc3, the engine says the best continuation: is Qf5 (declining the sac but standing right beside Ron), Rxf5 (Hermione takes f5), and then Nxf5 (white knight takes Hermione) ! (referred in (5.2))
4.3.2. An alternative to Hermione takes f5 is: Bc5+ (Harry c5+). The 2 best moves here are Rxf5+ (Hermione takes f5+), and Bxe7 (Harry takes e7), but Hermione seems to die in both of them. The 3rd best move Rf3+ (again, the non-Hermione rook) I think saves Hermione...or maybe not.
4.3.3. Analysing the Qf5 line further makes me think white isn't necessarily playing to win (I mean the original Qxc3 is suboptimal) but more playing to get one of the intruders to be sacrificed or something.
Both the bishop and rook (Harry and Hermione) were in danger twice
5.1. Harry: Yes. Immediately before Ron checks king at end, Harry, as the dark square black bishop, is in line of sight of white queen.
5.2. Hermione: Seems possible. There's no indication for Hermione in danger, but I don't see why there can't be a game like this. Hermione is of course in implicit danger (see (4.3.1)).
The second black knight (Ron) took the most white pieces
- When playing a game, I don't know of anything where people keep track of which piece captured the most pieces. This sounds something like determining which piece is like the 'MVP', like the online pokemon TCG or csgo. I don't think this is a chess thing. But I guess it seems plausible.
The same black bishop (Harry) went 4 spaces to the right in the opening
- 7.1. Harry as the dark square black bishop could've done Bb4, like in Nimzo-Indian Defence, but Nimzo-Indian Defence is a response to 1.d4 and not 1.e4. Perhaps there's some variation of Scandinavian that involves Bb4. (Now looking it up...) Ah, found something. According to this, there's something called 'Scandinavian, Anderssen counter-attack orthodox attack' that goes
- e4 d5 2. exd5 Qxd5 3. Nc3 Qa5 4. d4 e5 5. dxe5 Bb4 6. Bd2 Nc6 7. Nf3
[White "Some 1.e4 player"]
[Black "Some Scandinavian player"]
1. e4 d5 2. exd5 Qxd5 3. Nc3 Qa5 4. d4 e5 5. dxe5 Bb4 6. Bd2 Nc6 7. Nf3
Also, notice that before capturing the queen, Harry is along the same diagonal of b4 and Harry's starting square (f8). But actually 4 spaces to the right is a pretty common thing I think like Ruy Lopez. (I quit chess for chess960 a long time ago, so ask someone else.)
- 7.2. I think this is a lot easier if this means Bc5 instead of Bb4 (w/c might be what is meant based on (3), but I guess I'm wrong because there's no move Bc5 in the same link above or in wiki.
Update: Does Magnus Carlsen Remember the Position in Harry Potter Movie?
Here, Carlsen claims that the moves Qc3 and Nc6 were played. Not sure either of those moves were played, but in the silman position Qxd3 (not Qd3 and not Qc3) and Nh3 (not Nh6) were played.