AC and eddy.

Usually, maintaining eddy currents needs energy to overcome the effects of resistive dissipation, and a static magnetic field (the one you can get from a static magnet or DC arrangement) can't provide any. But if the material (I mean a metallic one) were cold enough to achieve superconduction, you could induce currents with DC or with a static magnet (Just google "superconducting levitation", for an example). Superconduction doesn't mean extremely low resistance, but no resistance at all.

Eddy current is created due to purely transformer action to make it simple .

When there is a conductive element in a time varying magnetic field, a current flows (or an emf is created in the direction) that resists the variation in the field.

Obviously DC can not create this time varying magnetic field.

As far as the superconductor is concerned, again it is the eddy that is cerated when the magnet is brought in (the mechanical velocity of course make the magnetic field to change in a time varying fashion) and hence a opposing current is generated.

Even in a non superconductor a spike is created in switch on, which almost instantly dies.

For Electromegnetics and details - refer Maxwells equations

You can if you pulse the DC.

Why must it be pulsed?

I refer you to sb's answer citing Maxwell's equations, but pulsing DC on and off simulates AC in that it induces an eddy current in a nearby conductor (usually planar) which produces an orthogonal magnetic field which subsequently collapses after the DC is turned off but then reappears when the DC is turned on again. The electromagnetic, resonant frequency of the process will be the pulse frequency that produces the highest eddy currents.