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However, the key_5 block remained complete mystery, so I really let loose with in-depth analysis, but to no avail. Eventually I brushed it off as something that is there just for fun or to mess with player.
Seeing that you were interested in detailed analysis I felt compelled to gather and format my notes into cohesive text. I added it to the guide, as it is too lengthy for the comments.
If you thought that puzzle analysis was pretty deep, this one will make me look like a total nutcase, heh heh.
key_5 is part of a completely separate thing, but it has nothing to do with this puzzle. The minesweeper board shows a picture of a (pixelated) lemon if you fill in all the prime numbers, which is not necessary to know and also not particularly helpful. The red colored numbers on the board is a red herring.
I think it's really cool that you put so much effort into trying to solve this, which is why I've pretty much just given the answer here. I hope you at least had some fun trying to solve it, even though the real solution was a bit of a mess.
Like in the case of this puzzle, the incomplete numbers on the hint could have alluded to size of numbers you're after. I.e. (Assume that slots in parenthesis have corresponding hint letters written above them):
([ ][ ]) [1][0][6] ([ ][ ]) [3][4]
This way you can reinforce mid-way that you got the right numbers from the puzzle and you're on the right way. It also works in reverse: you can guess if you're decoding stairs right from slots required.