Well the "density doesn't change" thing is a bit weird. Technically density does change if mass does, since density is just mass divided by volume.
When the characters in the books (and that annotation) talk about density not changing, they mean that the body doesn't become more resilient or resistant to being punctured as you'd expect a more dense object to.
Which I think would have its own weird physics implications because of "Newton's impact depth approximation" [ https://en.wikipedia.org/wiki/Impact_depth ] - density itself, not just structural strength, affects how deep a high-speed projectile goes into a target, since there is more mass to be pushed out of the way (transfer momentum to).
EDIT: Flight would be hard (astronauts in the space station can't really fly/swim through air well), and artificial wings or some such would be too heavy. (Assuming Wax's volume is about 0.1 cubic meters, which is likely high, to be neutrally buoyant in air he would have to weigh about 120 grams. Even clothes would weigh him down.)