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RolfAndreassen comments on Ask an experimental physicist - Less Wrong
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They are not silly questions, I asked them myself (at least the one about the Sun) when I was a student. However, it seems army1987 got there before I did. So, yep, when converting from mass-energy to kinetic energy, the total bending of spacetime doesn't change. Then the photon heads out of the solar system, ever-so-slightly changing the orbits of the planets.
As for magnets, the energy is stored either in their internal structure, ie the domains in a classic iron magnet; or in the magnetic field density. I think these are equivalent formulations. An interesting experiment would be to make a magnet move a lot of stuff and see if it got weaker over time, as this theory predicts.
If you're not thinking of moving a lot of stuff at once, every time you pull a piece of the stuff back off the magnet where it was before you're returning energy back to the system, so the energy needn't eventually be exhausted. (Though I guess it still eventually be if the system is at a non-zero temperature, because in each cycle some of the energy could be wasted as heat.)