Re: Where does the energy for gyroscopic precession come from?

You are correct in that conservation of energy is not generally considered in discussions about gyroscopes. A more complete discussion than usually found is given by R. Jakeways in an article that can be found on-line at http://www.iop.org/EJ/article/0031-9120/3/2/307/pev3i2p79.pdf?request-id=2a6d7b1d-29a7-43f9-9695-f952e2500d8f
In a nutshell, conservation of energy does indeed require a small vertical motion. In ends up, however, not being a single drop. Instead, the gyroscope undergoes oscillations vertically and horizontally (with an average horizontal precession).

Given the oscillatory motions, a track of the sort that you describe would have much more complicated interactions with the gyroscope (collisions). The only way to avoid the initial oscillations would be to start the gyroscope at the appropriate initial angle, and start it off with its precessional motion. In that situation, the track would be initially supporting the gyroscope. There would be no net torque on it, and whatever motion you gave it (spin + precession) would remain unchanged.