Jalt runs to where Tangle was located and looks up to the noise he heard. With his bonus action, Jalt will cast Shield of Faith on Tangle. He then will cast Sacred Flame on the dragon rider if he can  (Dex 15 save) for 5

Tangle struggles to climb back up on the dragon, trying to get a look at how the rider is sat. Are they strapped on somehow?

Move: try to swing up onto the back again (assuming Acrobatics at Disadvantage again?) 27
Bonus Action: Eye for Detail to make a Perception check to notice some small detail, looking for a way to unseat the rider, or any other detail about the rider or what they might be carrying. 27

... pausing mid-turn here to determine what Action to take depending on what happens with these two (does she stab, jump off, slice a strap?) 

OOC > Welp... "Dice rolls not yet available for this section" strikes again. I'll just recreate the post, I guess, and hope for the best?

Tangle struggles to climb back up on the dragon, trying to get a look at how the rider is sat. Are they strapped on somehow?

Move: try to swing up onto the back again (assuming Acrobatics at Disadvantage again?) 13 
Bonus Action: Eye for Detail to make a Perception check to notice some small detail, looking for a way to unseat the rider, or any other detail about the rider or what they might be carrying. 27 

... pausing mid-turn here to determine what Action to take depending on what happens with these two (does she stab, jump off, slice a strap?) 

Tangle, you manage to maintain your grip on the dragon so far, but don't succeed in climbing back up onto its back. You dangle.  You are a dangling Tangle, so to speak.

The elf is not wearing any kind of armour that you can see, just robes. He turns to look down at you, and there's something vaguely familiar about his face, though you are certain you've never seen him before. He's seated directly upon the dragon's back; he seems to be a skilled rider, but there's nothing preventing you from trying to knock him off - you think you have a decent chance.

What will you do?

Tangle will attempt to sheath her sword... while dangling... if she can, she will try to pull him off, into the trees. If she cannot sheath her sword in her scabbard, she will try sheathe her sword in him.

She can sheathe her sword while dangling, yes.

To pull him off the dragon's back and fall down, Tangle will need to make a shove attack, with disadvantage.

OOC > Shove is what Tiggers love best!

Tangle sheathes her sabre and reaches for the rider to try to pull him off!

Shove (at Disadvantage, because dragon, sky, dangling, etc.): Athletics 9  

Acrobatics 8

Tangle, you shove the elf, and he looks for a moment as though he will maintain his balance... and then the dragon swerves a little to the left and the rider fails to anticipate. He tumbles off the back, and falls through the air, landing somewhere below - you're not sure where. 

Is Tangle doing anything else on her turn?

 Tangle drops and lands in a thick area of leaves between two branches; she is light enough not to fall immediately.  Looking over, she sees the elf has also landed with some grace, and is looking back at her with an expression of fury on his face.

The dragon is up next. The dragon flies in a tight arc, as it wasn't travelling so fast yet, and manages to grab the rider from the treetops, though it uses a full action to do so. It ends its turn in that space, ten feet away from Tangle, ready to fly off, or so it would seem.

Soar is up. Soar doesn't see anything, but has heard crashing sounds coming from the north and up somewhere.

Soar moves 30 feet toward the crashing sounds. Does she see anything?

She sees movement in the trees above and a little further ahead, yes.  And a shadow, as the green dragon swoops around and back in, to grab something from the treetops.

Option 1:

If Soar can see the dragon, she'll immediately cast hex on it and fire off an eldritch blast at it...

Beam 1: 14 Damage: 5 plus 6 necrotic from hex
Beam 2: 21 Damage: 12 plus 5 necrotic from hex

Option 2:

If Soar can actually see the dragon grabbing something, AND it's within 60 feet of her, she'll cast command and tell it to "DROP"

Option 3:

If Soar can only see shadow, and can't actually see anything, or anyone, she calls out quietly, "Tangle?" and casts Armor of Agathys on herself at second level.

Option 4: 

If Soar is somehow able to get within melee of something hostile (still with an action), she will try to hit it with Talon.

Option 3 is the one that applies, so Soar casts Armor of Agathys on herself.

The dragon takes a legendary action, and beats its wings five times, each more mightily than the one before. Tangle, make a DEX save.

The dragon then flies off, forty feet to the north. Tangle can still see it, only just.

Round Three:
Bazavur
Rider
Jalt and Tangle
Dragon
Soar

Bazavur is up.

Tangle's Dex Save: 15

Bazavur moves towards the sounds as quickly as he can. Hoping he is able to aid his companions before it is too late. 

Tangle falls, and then falls some more, down to the ground.  She takes 14 and is prone.

Bazavur moves forward, in time to see Tangle tumble and crash onto the ground.

It's unclear what, if anything, the rider does.

Soar and Tangle are up. You don't see the dragon from where you are.

(14 bludgeoning damage from the fall, that means.)

Soar moves to Tangle, looking up into the trees, and assuming she sees nothing, lays on hands for 14hp.

If she does see something, she'll fire an eldritch blast at them. 

"What happened?!" she stage whispers.

"Graaaahhh!" Tangle screams in frustration and leaps up, ripping her longbow off her back and racing through the woods in the direction the dragon went, best she can tell.

"Green dragon with a rider, met with Galin, sensed me, scooped him up and flew off!" she yells over her shoulder as she scrambles away.

Move: 15' to stand, 75' with Move, Action, and Cunning Action. She's not giving up until it is clear the dragon is out of reach.