Should these kind of problems be tagged chess?
As I described in the comments of the first question you linked to, I don't believe the chess tag is appropriate here. The question is not pertinent to the game of chess so much as to the board on which it is played. The question could just as easily have been about a checker board, but that doesn't imply that it should be tagged checkers.
The second question you linked to talks about a "pawn", but again it doesn't seem to be a "chess pawn" but rather a piece that has its own rules for moving.
Arshajii does make a good point. I believe that the chess must be kept for questions that actually deal with the rules of chess.
Also note that a chessboard is in fact a type of checkerboard (ref. Wikipedia), and in cases like these where it does not matter that the board is made specifically for chess, the more general term checkerboard (lit. "A checkered board") should be used instead.
I would say no. Those two problems doesn't actually involve the game of chess in any way, just an 8-by-8 checkerboard (which I called a "chessboard" but we can rename that tag if you want to).
More recently, I saw this question which actually involves the chess pieces and how they move, which had the chess tag removed. I put it back on before noticing that the last edit had been the removal of the tag. Should questions like those be tagged with chess?
Since everyone seemed to be in favor of changing the tag to include board, I've gone ahead and made the change to all questions currently tagged chess (none of them were actually just about chess). They're currently pending approval so please go help out with that!
I selected chessboard rather than checkerboard because we seemed split on that, and you want to focus on usability: if someone is trying to type "chess", they will see the "chessboard" tag autocomplete and use that instead. If we used "checkerboard" people would keep misusing chess and we'd never fix the problem.