Since we are talking about a standard game of chess (although with both players co-operating), we know that there are four pieces that cannot possibly make a capture in the series:
the two bishops on the wrong coloured squares
one of the kings (the other can be the last to capture)
the first piece that gets captured (cannot be any of the above).
This answer was a group effort between @Graylocke, @berkeleybross, and myself, with a little confirmation from @Enigma, so I'm more than happy to make this a community wiki answer (unless there are any objections or @Stiv wants the +2 rep for accepting XD).
The answers to the questions:
What you should have done is
I have a computer program for solving packing problems, and found a way to use it to solve this problem. One of the solutions it found is below:
Note that this is very close to the attempted solution in the question, as it differs only in the top left quadrant.
Modelling the problem in this way will not find all solutions (I'd need to allow other tile ...
Vepir has helped twice in this answer, first in spotting a mistake and then with an improvement in the number of moves. Please got upvote their answer too if you like this one.
Here is a position with
This position can easily be reached in
This is also (one would hope) not the intended answer, but I'll go with the garbagious
That would be because to my eye white can only win by
after which there are a lot of transformers, as
The sequence seems forced: white wins this way, and any deviation from black leads to being down a queen.
After this, the white king will step to b2, confining the ...
Looks like after
the black king has only bad options:
and finally, after
we have reached this position:
All the escape squares of the black knight are blocked, so it will fall, which brings us to..
This assessment seems accurate, since the white bishop can sacrifice itself to stop black's e-pawn if necessary, after which black is left with a pawn on the ...
First things first: let's check the divisibility.
There are 64 squares, and the knight is standing in one of them. That leaves 63 squares to cover, and each move covers 3 squares, so that seems to work out. That means, however, that we won't be able to create a closed loop, so every solution we find only solves the puzzle for the starting point and, by ...
requires the following 40 moves to move the king from a1 to b7:
As a bonus, the White king is alone (fits the 'solitaire' theme'), the position is legal, and it doesn't use promoted pieces. The pawn on a5 and the bishop on h7 aren't even necessary, so there's probably room for some optimization.
Stealing ideas from Sleafar's first solution, I got to 62 moves. So hopefully someone can better that with the magic 64 or more.
EDIT: Since 62 is beaten by Sleafar, I'll up my score to 70: Earlier I failed to realize a8 could just be a pawn; freeing up a bishop
Addendum #2: With @Retudin's suggestion of placing an additional Black pawn on the bank tank, to the 129-mover, and my idea of moving around a few pieces, we have here a new record of 135 moves, with the king starting in check. The solution is still inherently the same, so there is no need to put it in.
Further Edit: Since there is no need for a unique ...