Crazy Aaron's Thinking Putty meets all four criteria: https://puttyworld.com.

Also I have one of these interlocking cube things and they are irresistable: http://www.notcot.com/archives/2012/08/3d-printed-cube-of-dodecahedro.php. Mine is my longest-lasting fidget toy so far. It's too light (in nylon), though; maybe try a heavier material.