The Green Dragon's breath is described as:
Poison Breath (Recharge 5-6). The dragon exhales poisonous
gas in a 90-foot cone. Each creature in that area must make
a DC 22 Constitution saving throw, taking 77 (22d6) poison
damage on a failed save, or half as much damage on a
successful one.
It is "poisonous gas" the Dwarf gets advantage on the save.
Poison. Venomous stings and the toxic gas of a green
dragon’s breath deal poison damage.
It is "poison damage" the Dwarf takes half damage (effectively 1/4 if saved 1/2 if not).
With respect to your comments on protection from poison, I think you are splitting hairs: it is poison so advantage is given on the save; the fact that the effect is to do poison damage rather than impose the poisoned condition is irrelevant to the save.
That said, the DM is the sole authority on rulings so, check with your DM.
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
A dragon's breath weapon takes an average of 18 seconds to recharge.
Most dragon breath weapons contain the "Recharge 5-6" notation, which means:
So each round, there is a 1 in 3 chance the weapon recharges, and so with six second rounds, you can expect an average recharge time of 18 seconds. Thus, it would be highly unusual for a dragon to not recharge its breath in 60 seconds (about a 1.7% chance of not recharging 10 rounds in a row).
The DM decides how the dragon behaves.
You ask:
And the answer to this is, well, nothing, except that the DM doesn't want the dragon to do this. You have identified a tactical advantage that dragons have when battling in the open air. The range of an adult dragon's breath weapon keeps them out of range of melee opportunity attacks, so readied ranged attacks and spells are the only option.
But the DM needs to consider "how can I make this encounter fun" before considering "how can I beat the players". Unless "DM vs. Players" is agreed to prior to starting play, D&D is by default a cooperative game, not "DM vs. Players":
Surprising the players with an unwinnable encounter is usually a bad idea unless you've got some other tricks up your sleeve to keep the players engaged and interested. Getting hit-and-run over and over by a dragon isn't going to be particularly fun, but a head-to-head struggle in the dirt might be, and will still be a difficult encounter.
I use "open air dragons" for show and tell, not the actual encounter.
What I gave above is the "metagame" reason for dragons not to engage in "open air" combat with the players. But I have been able to rationalize it within the narrative of my own campaigns just fine. Unless the party has done something to severely upset a particular dragon, the dragon probably doesn't have any reason to just engage the party outside of their lair. Depending on the lair features and your use of lair actions, a fight with a dragon can be just as difficult, if not more difficult than the situation you present. But there is one key difference: the difficulty provided by the lair and lair actions is typically going to be far more interesting than hit-and-run open air tactics. There is just a lot more going on in the lair that won't be quite as frustrating as getting kited in a field.
And since you're going for something true to a dragon's preferred tactics, they're going to prefer fighting in their lair anyway:
Which brings to what I mean by "show and tell". When I use a dragon in my stories, if there is going to be a fight, it's going to occur in the lair. When the dragon is seen flying around in the sky, it's sending a message to the party. "Hey guys, there's a dragon here, so watch out for dragon related plot devices". And then when it comes time to personally introduce our dragon, I will use an open air encounter to give the party an idea of what they are up against, without intending to carry on the fight. So dragon campaign arcs, for me, usually have three actual appearances of the dragon:
1. Observe that a dragon exists
The party sees a notice on the bulletin board at the local tavern, "Adventurers Wanted Up North", asking them to meet with the governor of Up-North-Land. While travelling, once the party is near their destination, they will observe the dragon. "As you break the crest of the last low mountain, you peer down into a snowy valley. Across the valley, you catch a glimpse of something catching the late afternoon sun — something you have heard stories about, but have never seen with your own eyes. The glistening outline of a silver or white dragon circles in the sky above the far mountains across the valley."
2. Taste the power of the dragon
You never want your party to try to take on the dragon too early. You don't want them to force you into a situation where you have to kill them or make the dragon do dumb stuff. This is where this second appearance of the dragon comes in handy. Find an opportunity in your story to put the dragon's true power on display, before the party is committed to a fight with them. In this campaign with the white dragon, my players were tasked with stealing a powerful relic from an ancient temple that was guarded by the white dragon. But they didn't know it was guarded by the white dragon. They had the bright idea to walk up to the front door and knock, so on the way, I had them meet the white dragon. I had the dragon swoop down while they were navigating across the tundra to the temple and blast them with Cold Breath, then fly off in the direction of the temple. They got the memo. After regrouping and doing some more research, they learned of a secret underground entrance to the temple and were able to get in and out with alerting the dragon.
3. Actually fight the dragon.
Eventually, this campaign came down to a showdown with the dragon in the dragon's lair. At this point, they had seen the dragon a few other times, but the one brief encounter where they got blasted with cold breath was enough to be sure they were prepared.