I successfully solved 7/8 problems and I believe the one left has a big flaw. Also, one solved has a minor flaw. I might be mistaken so the public (and the author) would probably correct me.
Now, as you noticed, the problems are unsolvable if you see them as chess problems. However, let's add a little tweak to chess rules:
If a group of figures is surrounded by enemy's figures from all sides (enemies hold a close chain, just like in Go game), then this group entirely burns in flames, leaving no trace. This takes place the same turn, before the turn's end.
If you read the previous spoiler I highly advice you to try to solve these problems once again. They're extraordinarily good.
Now let's get started:
One
I might have had to mention it - a burned king still counts as a mate. Next!
Two
The burn is used to clear a way for long-range fighters.
Three
Nothing new, really. Now we're in for two-turners.
Four
What's new here - the king can't go on the queen's place because that would burn him immediately (technically, a self-mate). That would be much more sacrificial than killing entire team, though.
Five
The best situation by far. Not only bishop sacrifices himself to clear the way, the king has to go up close to a rook to burn it. I even made you a meme about that (you still reading, huh?)
I leave six for now. Seven.
And eight. Both are pretty self-explanatory.
Flaws (?)
The solution of six as I got it:
What I mean here is - the f2 pawn is a hippy and it tries to convince g2 knight to be hippy too, but he finally decides to not and they burn the king.
Also that pink spot means the spot where the king would die, burned by the rook.
Why did I make such a strange decision? Well, shoot, if you have a better one. This decision was provided by the letter hint. Here is the hint I got, with purple letters taken from the other puzzles, and red - from puzzle six with this particular solution.
Also, if I didn't assume hippies - this would be a two-turn mate with indefinite end, not a three-turner.
SO THE ANSWER SEEMS TO BE
KNIGHT TO BE ONE WITH NATURE
Coincidentially, that's exactly what the f2 pawn said to the g2 knight.
Also Flaws
In puzzle 4, the last turn could be rot13(Ep7), which would foil the final message. Probably, replace with a knight?