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.

There are two systems that govern stealth: the Hide action and Travel Pace

Hiding

If you are scouting a specific enemy, you should use the Hide mechanics which involve using the Hide action (to provide a DC for any potential onlookers). A controlled mount, such as your horse, cannot use the Hide action since the only actions it can take are Dash, Disengage, and Dodge.

a controlled mount ... moves as you direct it, and it has only three action options: Dash, Disengage, and Dodge.

If you are not mounting the horse at the time, there is no reason you couldn't try to convince it to use the Hide action, see below for some idea how the Animal Handling would work. This could be made easier with spells such as beast bond or speak with animals.

Travel Pace

If you are simply scouting ahead of your party while travelling in an attempt to be prepared to warn your party of threats ahead of time, use the Travel Pace rules

$$ \text{Travel Pace and Effects} \\ \begin{array}{|c|c|} \hline \textbf{Pace} & \textbf{[...]} & \textbf{Effects} \\ \hline \text{Fast} & \text{[...]} & \text{−5 penalty to passive Wisdom (Perception) scores} \\ \hline \text{Normal} & \text{[...]} & \text{-} \\ \hline \text{Slow} & \text{[...]} & \text{Able to use stealth} \\ \hline \end{array} $$

Since creature's travelling at a Slow pace can use stealth, we look to the rules on Dexterity ability checks where it says:

Make a Dexterity (Stealth) check when you attempt to conceal yourself from enemies ... or sneak up on someone without being seen or heard.

As such, all creatures involved in the scouting (albeit at a Slow travel pace unless your mount is your Ranger's Companion) could use a Dexterity (Stealth) check to avoid notice. This means your party would also have to travel at this pace or they would catch up to you fairly quickly. To convince your horse to move stealthily would involve this section on Animal Handling:

When there is any question whether you can calm down a domesticated animal, keep a mount from getting spooked, or intuit an animal’s intentions, the GM might call for a Wisdom (Animal Handling) check. You also make a Wisdom (Animal Handling) check to control your mount when you attempt a risky maneuver.

Animal Handling

Nothing specifically allows for the instruction of your mount to move stealthily but if adequately trained, it would likely be achievable (I would probably consider exotic mounts proficient in stealth such as the tiger to already be somwhat "trained" in this way). For a creature untrained, it is likely almost impossible. Your GM could assign a relevant DC for Animal Handling with these rules, basing the difficulty on the level of training for stealth the mount has received:

$$ \text{Typical Difficulty Classes} \\ \begin{array}{|c|c|} \hline \textbf{Task Difficulty} & \textbf{DC} \\ \hline \text{Very Easy} & \text{5} \\ \hline \text{Easy} & \text{10} \\ \hline \text{Medium} & \text{15} \\ \hline \text {Hard} & \text{20} \\ \hline \text{Very Hard} & \text{25} \\ \hline \text{Nearly Impossible} & \text{30} \\ \hline \end{array} $$

I would rule that a mount with no training would result in a "Nearly Impossible" DC and move up the table for each level of training or comparable experience. I would use the Downtime rules for Training as a guide (see below, emphasis mine). Perhaps with each series of training requiring 3 successful Animal Handling checks before 3 failures (checks made periodically with 1/5 the total time and then are no longer needed after 3 successes, DC based on the level it has before training), you improve the mount's understanding of particular situations for stealthy movement. I would allow proficiency in stealth to give the mount a starting level of Medium. Once it is trained beyond Easy, I would stop requiring checks for this activity.

Training

You can spend time between adventures learning a new language or training with a set of tools. Your GM might allow additional training options... The GM determines how long it takes, and whether one or more ability checks are required. The training ... costs 1 gp per day.