2)Gravitional assists that space probes like Voyager 2 used to gain speed actually slow a planets rotation ever so slightly.
Exactly.
Without making a judgement on this technology in specific for a device of this nature to work it has to get the energy from somewhere.
If it does do what it claims it is most likely capturing some of the earth's magnetic field. If it does so and it gets popular and take enough eventually it will either have an effect on the earth's weather or its rotations(imagine a 30 hour day because the earth slowed!) or its core temp or some thing tied into the earth's magnetic core.