Would a white dwarf have enough gravity to produce a visible light-distortion?
Not really, not enough to bother trying to model it in SpaceEngine. Just think, neutron stars are one thousand times smaller, and a few times more massive. The difference in the noticeable distortion between the two would be staggering, much greater than the difference in noticeable distortion between the Sun and a white dwarf.
Also, can a neutron star, lets say in a binary system, have enough matter fall on it that it can evolve into a black hole?
Perhaps. But such an event would take place over astronomical timescales, and so wouldn't make sense to model in SpaceEngine until such things as stellar evolution and other long term changes are implemented.