my watch needs a 21mm strap, should I go up or down, if I choose this band?

The manufacturer might have a different answer, but a millimeter is a lot to be off by. My strap (the leather part) fits snugly into the opening of the watch. If it were 1 mm bigger, it wouldn’t go in. If it were 1 mm smaller, the leather would fit but I’d worry the metal pin wouldn’t expand enough to engage the watch. My advice is to find a manufacturer that offers odd sized straps.

I have a casio watch that has a 23.7mm band. Will the 24mm option work?

It would be pretty close. The one I have has a a little gap between the band and watch. You can always return it if it doesn't fit.

Does it work for fossil watches?

It works with all watches once there are no special parts to the watch

The withings watch model HWA03B is 36mm or 40mm?

bofore ordering the watch band Look at your watch piece. Where the strap will fit into between the post measure the inside in mm. Then order the next mm size for secure fit of spring to expand. hope this helps.