How close to the king are they covering? .0000001 units? The king can move .00000009 units. You can always make a smaller real number.

In theory yes. In practice, python uses floating point numbers for decimals so there is actually a minimum possible move and a finite number of allowed decimals.

Also the game only has a resolution of 640x640 pixels, and you cant move the mouse a half pixel. But anyways, stalemate basically requires very precise positioning by the attacker, so it is VERY unlikely to actually happen, especially since they are trying to avoid stalemate.

The king can put himself in check too, so the defender would have to cooperatively allow checkmate (instead of capturing the other pieces)

Not if the distance is zero, and that’s mathematically doable and possible in IEEE floats, certainly for the two rooks or two bishops version.

> Talking of bishops: should this game require bishops to stay on squares of their starting colour?

Given how captured work and how you'd probably define "on a square" with white a bit of freedom (perhaps the center of the piece has to be within the square?) I doubt it'd matter much either way.