there's the king gambit stenitz variation (1.e4 e5 2.f4 exf4 3.d4!?/?! Qh4+) where white king plays the whole game on the run. it's a semi respectable line, there is an IM who played the line a few times against equal opposition is serious tournament games. In theory it should be better for the second player but probably is not that easy to prove.
Commander King Openings
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There are some interesting castles in shogi (a variation of chess) which has kings on the fourth rank.
shogi (japanese chess) is not simply a variation of chess, it's an equally ancient and closely related game :) In fact it might be that chess is a vatiation of shogi :D
I only know the basic rules, i have been told that the game is tactically wilder and more difficult that "our" chess, but in a certain sense less positionally demending since pawns can eat the piece in fromt of them, meaning that you never have a fixed pawn structure.
there's the bongcloud.
most openings which feature something like this will have a refutation or just are considered simply bad and unnamed
Here's an idea of what might happen if the commander runs along ahead of his army:
Your first one was a little better than your second one.
Actually there are middle game postions where the king takes an active part in an aggresive and supportive way with the queens still on the board but
such situations are fairly rare.
Seriously though, the Black side of the Fried Liver might be one of the better examples. However, White seems to be close to winning with the Lolli attack so few people venture 5...Nxd5 these days.
I don't think commander king openings are good, but here is a game where my king was forced to run and I ended up winning:
I find the idea of the King as a commander/supporter of your chess troops fascinating. A bit like this:
Are there any openings which puts the king in an aggressive / supportive position in the middle game preferably without trading queens early?