Should I recapture the queen with my king or knight in the Scotch Game? I watched the following video on the Scotch Game by NM Dereque Kelley, and he clearly points out that 3...d6 is not a good move on Black's part because it causes all sorts of problems, especially coming from 4. Bb5 pinning the knight.  But if Black does play 3...d6 and 4. dxe5 dxe5 5. Qxd8+ follows, we arrive at a position that I have been in many times as White and as Black.  As Black, I normally would play 5...Nxd8, but in the video, Dereque says that this is not an option because it leaves the e5 pawn hanging, so Black must capture with 5...Kxd8 and give up castling rights.  Is this really the best move given that castling is a fundamental part of the opening?  I understand that 5...Nxd8 does leave the e5 pawn hanging, but what is more important here, holding on to the e5 pawn or maintaining the ability to castle?
As you say, each of the options 5. ... Nxd8 and 5. ... Kxd8 has its own drawbacks: the former drops a pawn and the latter gives up castling rights. In this situation Kelley is right that losing the pawn is significantly worse than giving up the right to castle.

In a typical middlegame that features a lot of firepower on board for both sides, a king stuck in the center of the board can be in a great deal of danger of facing a mating attack. But in this case, the king gets stuck in the center only after the queens come off the board, which means Black has much less to worry about in terms of facing an attack. This queenless middlegame is not so far removed from an endgame, in which centralization of the king is generally a plus rather than a minus. Now there are still a lot of pieces on the board, so it's not a walk in the park for the Black king; I'm just saying it's not as dire as things can be with the ladies still around.

This particular position still has certain problems for Black (which is one reason why 3. ... d6 isn't the best choice); for instance, the king having been pulled to d8 means that the f-pawn is undefended, so with something like 6. Bc4 White develops the bishop, and the Black either must waste a tempo moving the king back to defend the f-pawn, or play 6. ... Be6 to defend the pawn, allowing White the chance to give Black doubled, isolated pawns by 7. Bxe6 fxe6. That's not the end of the world, and in this kind of position those pawns do at least help control some central squares. The main point: Black will probably end up with some structural problems, or lag behind in development for a bit, but this is better than the alternative.

Losing a pawn isn't the end of the world either, it's true, but here it's much closer. After 5. ... Nxd8 6. Nxe5, White is a clear pawn up without any compensating factors in Black's favor; in fact White has a lead in development as well. This is something Black definitely shouldn't allow, especially since we really are getting toward the realm of the endgame, where having an extra pawn is of crucial importance. Going this route puts White in the enviable situation of "playing for two results," meaning that from here either a White win or a draw are nearly certain, with a loss for White being very unlikely (barring a big mistake later of course). That's one of the worst things Black can do to herself in the opening.