Developing the king before developing most of your pieces I saw this opening for the first time and was amazed by the position of the white king.
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1. e4 c6 2. d4 d5 3. e5 Bf5 4. h4 h6 5. g4 Bd7 6. h5 e6 7. f4 c5 8. c3 Nc6 9. Nf3 Qb6 10. Kf2 O-O-O 11. Kg3

In this position, white made one knight move, two king moves, and no moves with the other pieces. Usually, we should castle early and hide the king in a safe corner. In this opening, white does the opposite by moving his king to g3 in the opening stage before developing most of their pieces. Is it really a good idea? Are there any other openings that you develop your king early in the game?
Don't try this at home. ;-)

First of all, a simple LiChess search gives this is B12 Caro-Kann Defense: Advance Variation, Tal Variation, i.e. it has even a name and is legit.
Second, 8 games listed, with IMs playing it, so it's still rare.
Third, standard Black move is Kb8. He doesn't even try to immediately throw everything except the kitchen sink at the white king!
And that is because the king is rather safe there. Usually you now would try the can opener g6 or f6, but it's not so easy as the king side of Black is underdeveloped and White can keep the relevant files closed. Black can't simply pile up major force, and already Steinitz said you need a force plus to storm the defense.
This is caused by the large space advantage of White at the kingside.
A concrete advantage of wK on g3 is that it does something useful (protecting g4) and avoids something annoying (tactics on e1, pins on g1...).
Note that the center is totally barricaded, otherwise it would be suicide.
Summing up, yes, in this concrete position (modern chess is hyperconcrete) it might be a good idea.

Concerning your last question (are there other variants?), this could be easily answered by a search in a Megabase. Steinitz Gambit of course comes to mind immediately.