If the iron ball manages to induce a current in the coil then yes, the ball would slow down.
I think your question about where the energy comes from is badly phrased - the slowing ball doesn't provide the energy. The energy comes from a magnetic flux cutting a conductor, the prime mover (in this case the movement of the ball) provides the power, the flux cutting provides the method of power conversion, in your example, movement into electrical energy. If 'work is done' then the ball's movement must change, ie slow down.
I think that your physics teacher is correct but is answering you generally - you would find that the movement of an iron ball past a coil would be unlikely to induce current into the coil, it may induce an e.m.f. from its remnant magnetism, but would not have sufficient flux density to generate (in)significant power.
Finally, a ball orbiting inside a large coil isn't going to cut the conductor with its flux, your motion is wrong, so you will not generate here either. Even if your motion was correct, power used would slow the ball and it would lose its orbit and fall back to earth.
Type flux induced emf into Google and you will find a bunch of useful topics on Faraday's and Lenz's laws.
(
http://physics.bu.edu/~duffy/PY106/InducedEMF.html
was quite readable)