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.

Yes, movement may continue [at a penalty] when special movement speed has run out

You've definitely called out the correct section on movement speeds found in the book, but it's worth noting that in the time since the original printing of the PHB, there was errata added to make this kind of movement more explicitly clear:

Climbing, Swimming, and Crawling

Each foot of movement costs 1 extra foot (2 extra feet in difficult terrain) when you're climbing, swimming, or crawling. You ignore this extra cost if you have a climbing speed and use it to climb, or a swimming speed and use it to swim. At the DM's option, climbing a slippery vertical surface or one with few handholds requires a successful Strength (Athletics) check. Similarly, gaining any distance in rough water might require a successful Strength (Athletics) check.

—Movement, Player's Handbook, pg. 182 (with 2018 Errata)

So what we see here is that the movement cost only applies so long as the creature does not use their movement speed for that particular type of movement. It doesn't say that you're unable to use your walking speed simply because you have a climbing speed.

So if you're using your climbing speed, you'd use 1 foot of movement for each foot you move. Then, if you run out of climbing speed, you can keep going at a rate of 2 feet of movement for each foot you move. As a result, your third option, moving 15 feet with climbing speed, then moving 25 / 2 feet with their remaining walking speed is correct; though it should be noted that Rules-as-Written, there's no requirement to round distances, since 5th Edition does not "snap to a grid" by default. So it's perfectly legal for a character to move exactly 12.5 feet (or in this case, 27.5 feet in total) during their turn.