Braten - The description of Krykun's course on Chessable says he goes for "very rare or novel lines". Thus, it's not too surprising that I found very little coverage of the line you gave ending in 10...Ndxe5 and producing this position:



The position search functions on Modern Chess and Chess Publishing failed to find anything, and the only occurrence of the position on Chessable was in Krykun's course itself. A search of the Mega database finds only six games with the position. Black scores rather well (three wins, two losses, and draw), but it's difficult to draw any conclusions from such a small number of games. Clicking quickly through the games didn't reveal any clear preference for where to put the black king, although castling long didn't appear to be common. 

Since you spoke of "these kinds of positions", I can point to similar lines where more resources are available. For example, a rather more common variation of the French with a similar advance of the black g-pawn is 1.e4 e6 2.d4 d5 3.Nc3 Nf6 4.e5 Nfd7 5.Nf3 c5 6.dxc5 Nc6 7.Bf4 Nxc5 8.Bd3 Be7 9.0-0 g5, yielding this position: 



This line is covered in five Modern Chess databases:

French Defence - Top-Level Repertoire for Black (GM Ivan Cheparinov)
The Practical French Defense - Top-Level Repertoire for Black (IM Kushager Krishnater)
French Defense for Black - Play the Steinitz Variation (GM Michael Roiz and others)
French Defense for Black - The Practical Steinitz Variation (GM Michael Roiz and others)
Practical French for Black - Part 1 - (GM Felix Blohberger)

The first two databases only cover the line a couple moves beyond the diagrammed position. Play the Steinitz Variation by Roiz goes a bit deeper, but doesn't provide much instruction on how to play these sorts of positions. A couple games are cited, and these might be worth playing through. The Practical Steinitz Variation just repeats the analysis from Play the Steinitz Variation. Blohberger's database was just released this weekend. From the diagram, he continues with 10.Bg3 h5 11.h3 Bd7, indicating that Black will develop a dangerous kingside attack. However, he doesn't say anything about where Black should put the king.

The position also occurs in two Chessable courses:

Master the French Defense (NM Bryan Tillis)
Understanding the French: A Fighting Repertoire for Improvers (GM Jose Gonzalez)

Tillis' course gives a bit deeper coverage and a better explanation on how to play the position. He suggests play on the queenside with moves such as ...Qb6, ...Bd7, and ...Rc8, while adding that "Black's king will be safe in the locked center", which seemingly implies it will stay on e8, at least for a while. 

Instead of buying an entire course, it might be worthwhile to check out the single line associated with the diagram position using the Explore Variation feature on the Chessable site.

In the French section of Chess Publishing, John Watson mentions the position briefly in his notes to Druckenthaner-Drozdovskij, Hastings 2009, noting that White should play 9.h4 to avoid 9.0-0 g5, which suggests to me that he thinks Black is better after 9...g5. 

Another similar position arises from the Morozevich Variation of the Tarrasch French after 1.e4 e6 2.d4 d5 3.Nd2 Be7 4.Ngf3 Nf6 5.e5 Nfd7 6.Bd3 c5 7.c3 Nc6 8.0-0 g5:



Note that the position can also come about from the Universal Variation: 1.e4 e6 2.d4 d5 3.Nd2 Nf6 4. e5 Nfd7 5.Ngf3 c5 6.c3 Nc6 7.Bd3 Be7 8.0-0 g5.

Four Modern Chess opening databases include this position, but all of these give repertoires for White, which may be of limited use to someone looking for instruction on how to play the Black side of the position. On the other hand, Tillis' Master the French Defense on Chessable provides considerably more analysis. Some lines go beyond move 20, which means plans for the early middle game (including king placement) are presented. Note that Tillis' course uses the Universal Variation move order.

Two older books cover the position via the Morozevich move order: 

The Modern French: A Complete Guide for Black (GM Dejan Antic and IM Branimir Maksimovic, 2012)
French Defence Reloaded (GM Nikita Vitiugov, 2012)

Both books analyze the position deeply, well beyond move 20 in some lines. In doing so, they indicate how to handle the middle game, including where to put the king.

A more modern view of the position may be found in ChessBase Magazine #205 (2022) in an article on this line of the Morozevich variation by IM Christian Braun. He gives five deeply analyzed lines that end between moves 18 and 27.   

Chess Publishing has examined the position extensively, with about thirty games dating from 2001 to 2026, each annotated by Neil McDonald, Kevin Goh Wei Ming, or John Watson.

I think every game you play, or play through, will advance your understanding and confidence. I'm attracted by these chaotic positions and I think that is part of the appeal of the French Defense.

Breaking up white's center, as here, is a significant accomplishment. Black's king is currently safe behind a wall of pawns and with good piece development. White's king bishop is often a significant attacking piece and you may be able to trade it off.

It's true that you may not know where your king belongs. However, it's a plus that white does not know where your king is going.

I hope that helps.

Your final position could also be reached after the moves
1. e4 e6 2. d4 d5 3. Nc3 Nf6 4. e5 Nfd7 5. Nf3 c5 6. dc5 Nc6 7. Bf4 Be7 8. Bd3 g5 9. Bg3 g4 10. Nd2 Nde5.