As far as I can determine, there are no guidelines for exactly what this ability covers. However, I would assume that this is referring to physical indistinguishability (is that a word?) and therefore I would rule as the following:
1. Does an ordinary, completely harmless stalactite gain Unseen Attacker advantage when it attempts to throttle a character within its reach?
Yes. If a creature is unseen when it attacks, it has the advantage. Note that attacking will reveal it, so this will only apply to the first attack the creature makes.
2. Is it only visually indistinguishable, or should this also fool other senses?
This is tricky. On the other hand, it has to remain still for this ability to function, so sound is not an issue. I wouldn't assume the creature would have strong BO - this would invalidate the ability somewhat. However, if a player went up to it and touched it or sniffed at it, I would at the very least allow them to make a Perception or Investigation roll.
3. Should the PCs even get a chance to use Perception (either passive or as a check)?
As I mentioned in the previous point, if a player was to feel the creature or sniff at it from right up close, I'd probably allow Perception to apply. But if this ability was meant to interact with Perception, I'd assume it would say something like "While [the creature] remains motionless, it is indistinguishable from a normal [ostensibly harmless object] and gains a +X bonus to its Stealth check." Since it doesn't include anything to make it more difficult to succeed on a Perception check, my assumption is that it's not meant to be on the receiving end of Perception checks at all.
4. Once it's moved, should the stalactite need an action to become motionless again?
I would rule that the creature would have to make a Hide action to allow this ability to function. Note that this shouldn't really matter - if it's not in the vicinity of the players, it won't matter whether it needs an action or not. If it is in the vicinity of the players, I'd imagine they'd notice a stalactite shuffling along the ceiling and pay attention to where it stopped moving.
5. Should indistinguishability extend to magic (e.g. Detect Magic, Truesight)?
I would assume not. However, Detect Magic would only detect a creature with a magical effect on it, and this ability isn't an illusion that can be seen through with Truesight, it's a purely physical effect. Something like Detect Thoughts, on the other hand, should work fine.
The combat begins when the Darkmantle decides to attack. If the players don't notice the Darkmantle at the beginning of combat, they are surprised:
Any character or monster that doesn’t notice a threat is surprised.
So, thanks to False Appearance, the Darkmantle has a Surprise attack. Also, when the combat starts, the adventurers don't know that they're in a combat situation and so are not scanning the entire space all the time, unless previously stated (they are not combat-aware), so the Darkmantle remains hidden until its turn. However, all characters that are higher in the initiative order than the Darkmantle are no longer surprised at the end of their turn (or, better say, the "Surprise" property is no longer applied to them)! They are thus able to take reactions. For reference, this answer to another question (also linked in the comment of this question) breaks it down in full detail.
The Unseen Attacker advantage, however, depends on the DM's ruling, as the RAW are not specific about this.
As far as Darkmantles go, they could approach their target unnoticed. They don't have to fly, they can just drop down. One could reasonably fall unnoticed as easily as dropping a curtain on top of someone. Even if someone would notice the creature as it was falling, they'd only get enough time to look up and see the horrid array of teeth approaching swiftly.
If you still believe your Darkmantles are unable to automatically Batman your PC's, then a Stealth check is appropriate, and should be made during the Darkmantle's Move action. In either case, the Surprise Attack persists, it is only the Unseen Attacker advantage that is at stake.
Keep in mind that the target of the attack must be taken into account. If the Darkmantle attacks a character that was higher in the initiative order, the character can take reactions (like Tempest Cleric's Wrath of the Storm), regardless of whether the Darkmantle remained hidden during its movement or not. The very act of attacking reveals it:
If you are hidden—both unseen and unheard—when you make an attack, you give away your location when the attack hits or misses.
However, if the trigger for their reaction is before the attack, the Darkmantle's visibility has to be taken into account.
If the Darkmantle attacks a character that's lower in the initiative order, the character won't be able to react until the end of its own turn. So, even if his reactions trigger "on hit", he won't be able to use them. However, if the characters that went before the Darkmantle have a reaction that triggers "when an ally is attacked", they can take that reaction (again, because the attack reveals it).
There is only the final, special case to consider, where the Stealth check is only partially successful and one or more characters notice the Darkmantle as it moves in to strike. It is fully within their ability to warn the others of the threat, as they can talk out of turn (within reason). However, as it's the Darkmantle's movement that triggers the Stealth check, the players that were surprised at the moment of the attack still remain surprised. Whether the darkmantle retains its Unseen Attacker advantage or not is up to the DM and his opinion on how fast a character can react to a warning.
From a role-playing perspective, if the Darkmantle remains unseen the adventurers either didn't notice the attack until the moment it hit, or they noticed it too late to do anything about it (including removing the advantage).
Here's an example of how that could be acted out:
(A PC walks under the Darkmantle and it decides to attack.)
DM: As you explore the dark cave, you feel a slight chill up your spine... Roll Initiative!
Player: Why? Do we see anyone here?
DM: Not yet.
(Initiative order is set. )
DM (to the players at the top of the initiative order, before the Darkmantle): You are not aware of any threats and proceed as you were.
(The Darkmantle's turn comes. It makes its optional Stealth check... and remains hidden. It swoops down on its unsuspecting target and rolls its attack with advantage... success!)
DM: Alright... Aldore, your head is suddenly wrapped in a curtain of flesh and you feel piercing teeth gnawing at your cranium. You take XX damage!
(DM now describes the attacker, and the battle continues. The characters that haven't had their turns yet are still surprised. When all turns resolve, the combat continues as usual from the top of the initiative order.)
Now, if the creature missed the attack (either by lacking or despite the advantage) you could have it be because the character noticed it and dodged in the last second, or you could make the Darkmantle comically flop on the ground in the middle of the party. Your choice :)
Best Answer
The PCs would detect the Cloaker for what it is immediately as it is picked up. Presumably, only the topside of the Cloaker is camouflaged, while on the underside are its head, evil eyes and teeth. Here's the relevant text from the Cloaker's False Appearance trait:
Picking up the cloak would immediately expose at least some of its underside, revealing the Cloaker's more bite-y under parts. At that point, the Cloaker has achieved its goal as an ambush predator, it's gotten close enough to its prey and is going to surprise the poor PC.
If, somehow, the Cloaker weren't discovered and the PC puts it on, it's best to really think about the creature's motivations:
I recommend reading this TheAngryGM article (TRIGGER WARNING: swearing and jokingly belittling players) about encounters, setting up conflicts and creature motivations as further reading.