The challenge is in making a physical object rotate fast enough to produce the effect. The article says "the fastest rotation achievable by standard motors is of the order of 10 kHz", which is apparently too slow. The frequency of visible light, for comparison, is about 400–700 terahertz (THz).
Yes; in the same way you can influence gravity by a spinning mass, albeit we do not possess (by orders of magnitude) material or energy required to spin a mass fast enough to detect an effect. Spinning supermassive blackholes show a gravitation/time frame-dragging effect dependent on speed of spin. Showing it occurs with EM is amazing.
This is probably a really dumb question but could this effect be the cause of the discrepancy of the expected and observed rotational velocity of galaxies?