The answer is, I believe, less obvious than it first appears. The first thing to realise is that the moon already is moving away from the Earth for exactly this reason. The moon moves water around the Earth. This generates kinetic energy, which is dissipated into heat and eventually radiates away, and the overall effect is to make the moon move away from the Earth. (Actually I suspect that most of this effect is associated with squishing the Earth's rocky interior than moving the oceans around, but I'm not sure of the numbers. The oceans surely play a role though.)
The question then becomes, what happens if, instead of being dissipated by friction in the oceans, the energy generated by the tides is first turned into electricity by humans and then turned into heat?
A naïve first answer to this new question would be something along the lines of "nothing - there's a set amount of energy being generated, and it doesn't matter whether it's used up by humans or by tidal friction, so the moon will continue to recede from the Earth at the same rate that it already does". However, this doesn't take account of feedbacks. If you extract that much energy from the tides you're bound to change their behaviour in some way. I would guess that the tides would become shallower, resulting in less energy being turned to heat overall, meaning that the moon's rate of recession from the Earth would slow down slightly. But it isn't obvious, at least to me, and I suspect it's a problem that would require some fancy fluid dynamics in order to solve.
The question about rotation can be answered by considering the angular momentum of the Earth-moon system. If the rotation speeds of the moon and the Earth stayed constant, the system would gain angular momentum as the moon moved further away. But the total angular momentum of the Earth-moon system cannot change, so the Earth's rotation slows down in order to counteract this. The moon's rotation has already slowed down as much as it can do - this is why it always shows the same face to the Earth. In theory the Earth will eventually become locked to the moon in the same way, but I've no idea if this will happen before or after the sun expands and destroys them both.