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
Yes, That Description is RAW
That is the way mechanics happens in RAW. How you narrate that to make sense is up to you. Remember the rules have to be turn based to make it run and everything that happens in a round is happening inside 6 seconds and at roughly at the "same time".
Two Common Alternatives
There are two common alternatives I've seen:
Surprise Round - Borrowing from other system, including 3.5 and Pathfinder, some DMs I've seen hold on to a full surprise round. It would mean the surprised creature doesn't get its reaction back until after B has gone.
The action that starts surprise combat happens before the first round - The dagger is thrown and damage dealt, then initiative is rolled, and the creature has the surprised condition for the first round of combat. This is what I play, and I first saw it in how Matt Mercer of Critical Role DMs.
The players must have passed a stealth check against the character's perception. Note, that players don't know if they are really hidden, until they attempt to sneak by or attack. If they aren't stealthy as they think, I narrate something like, "You ready your dagger to throw, and X looks at you. Roll initiative."
I allow monsters to do it very occasionally as well (if players can, the NPCs can, too). It works well with narrative. I make the stealth check behind the screen, check the passive perception, or ask for a roll (roll is better here, as it lets the players feel I didn't do arbitrarily). Then, "You feel a sharp pain as you down you notice an arrow has sprouted from the joint in the armor. You look around to find the source. You are surprised to see X staring at you. Roll initiative." And if it fails to beat their perception check when a monster tries it, my players love it. "You see an X creeping in the forest, it appears to think you don't see him. What would you like to do?"
From comments, it has also been suggested that a common method is to give a initiative bonus to the surprise attacker who starts the combat, or to just let them go first in the initiative order. This works with the narrative and isn't as powerful as 1 or 2, and is less likely to have balance issues.
Game Designers and Initiative
There are a few other options for how initiative is done posted by game designer Mike Mearls to Unearthed Arcana back in 2017. In it he proposes initiative is calculated the other way, higher number goes last. Surprise adds 10 to the initiative and the creature can't take reactions until its turn. This isn't a common way, but it is very interesting.