[RPG] real use for the Medicine skill

Passive skills (especially perception, but anything really), are representative of a sort of "always on" ability. So, for instance, if you're walking in a forest and there is a goblin ambush, you see them if your passive perception beats their stealth score. However, if you're specifically looking for ambushes, you'll roll it actively.

So in the secret doors case, when you walk into a room, if your passive perception is high enough to spot the secret door, you notice it. If you're specifically looking for a secret door, then you roll an active check to try to notice (naturally, if you roll less than 10, you don't learn anything new).

Generally, the only time to roll an active perception check is if you are looking for something that your passive perception did not pick up. Naturally, in the course of play this happens fairly regularly. But the passive skills are designed to reflect a characters natural competence and should be used when they are sufficient, or there is no particular rush.

You can't treat the time you spend unconscious as a short rest.

Short Rest

A short rest is a period of downtime, at least 1 hour long, during which a character does nothing more strenuous than eating, drinking, reading, and tending to wounds.

There's a big difference between that and an hour spent unconscious because you almost died. I suspect that as far as the ability to regain hit points is concerned, the "tending to wounds" bit is the most important.

As you say, there is nothing in the rules that allows a Medicine check to restore hit points. In your capacity as the DM you are free to change the rules however you wish, but within the scope of the rules, only magic or resting can restore hit points.