[RPG] What happens when a creature clinging to a surface through the Spider Climb spell goes prone

The answer is in your bold letters.

The subject can climb and travel on vertical surfaces or even traverse ceilings as well as a spider does. The affected creature must have its hands free to climb in this manner. The subject gains a climb speed of 20 feet; furthermore, it need not make Climb checks to traverse a vertical or horizontal surface (even upside down).

The description says that it doesn't need to make Climb checks in order to traverse a vertical or horizontal surface. In that case, the check is to overcome the slippery effect of Grease spell. I would assume that it is the same as saying that a creature with walking speed needn't make any Balance checks to walk on an horizontal surface. But it still needs to make them in order to overcome obstacles. Grease is an obstacle, not a surface.

The spell calls for a Balance check, but given the specificity of the situation you describe, your GM could perfectly call for a Climb check instead. It seems reasonable.

As a general rule, the magical item continues to function, however, what happens to the wearer of the shoes is a different matter. Since there is no guidance on what happens to an unconscious creature who is standing on the wall, we will need to make some assumptions. And just because the shoes allow him to walk on walls and ceilings doesn't mean that our assumptions about gravity in the game world stop applying.

If he falls unconscious and is unable to control his body, then it stands to reason he will fall down. The magic item continues operating, but that doesn't mean he is magically stuck to the ceiling (or does it? see below...). I would rule the same for a spider on a wall or ceiling. With no mental faculties with which to maintain control of your body, you go limp and fall.

To be clear, this answer is based on my discretion as a DM, as there is no rule for what should happen here. If I were the DM in your scenario, the character would fall off the wall and down the well (though perhaps not drown -- that takes quite a bit of time in terms of rounds -- usually more than enough to mount a rescue attempt, anyway).

It would also be perfectly acceptable for you, as DM, to determine that Slippers of Spider Climbing do work as sticky pads which always work and never slip off the feet of the wearer, even when the wearer is unconscious and dangling upside-down from the ceiling over a pit of lava or what-have-you.