Perhaps you noticed the wikipedia article on Superconductor referenced the Meissner effect, which is "the complete ejection of magnetic field lines from the interior of the superconductor as it transitions into the superconducting state" (A material transitions to the superconducting state when it is cooled below a certain critical temperature). Induction doesnt take place in this situation as it would in any other. The article continues; "In a weak applied field, a superconductor "expels" nearly all magnetic flux. It does this by setting up electric currents near its surface. The magnetic field of these surface currents cancels the applied magnetic field". If you search superconductor in google images you see a lot of this:
This is a picture of a magnet hovering over a piece of superconducting material (which has been cooled below its critical temperature; notice the liquid nitrogen evaporating). The magnet hovers because of the Meissner effect. Magnetic flux from the hovering magnet is expelled from the superconducting material below it. Generally induction is thought of as an electromagnetic field influencing the electromagnetic properties of a material, but in this case the material has zero resistance. The magnetic field lines are directly responsible for the electrical currents in the superconducting material which balance them out.
So in a way you could say yes, the magnet alters the current in the superconductor, but it generates a current which induces a balancing magnetic field. Power is out of the picture because there is no resistance; the stationary magnetic field is enough to generate a current.
I dont think there is a theoretical limit to the amount of current that can be put through a superconducting material, but i also couldnt tell you what a practical limit might be. I think I sill have some information on possible explanations for practical limits, but I'd have to dig it out from my school stuff. Superconductivity has been studied for almost 100 years now but we're still missing a dynamical theory to explain the phenomenon. If you figure it out, you could probably get a nobel prize.
Hope this answered at least some of your questions, sorry I couldnt be more specific.
I wanted to post this to you sooner but I was accidentally banned because someone thought I posted spam...
Seeing as this is only my third post and my first one was in the welcome thing, Im guessing it had to do with one of my posts about opencourseware. Its rather interesting how quick they are around here to block someone out completely with no explanation or any way to contact administrators. I had to reply to the welcome email.