... I guess you could ask? Though they'd probably lie.
You could throw rocks at them one at a time, and see how many hit?

People wearing Full Plate are usually strong enough to be dangerous at your current age. When you're older you'll be able to Detect Magic at will, at which point you should go off of the most potent aura present, or count how many magic items they have between them in a group.

Dergaix would like to perform another Full Attack Power Attack, and then escape melee by taking a 5 Foot Step in the Up direction.

1d20 + 14 = 32, Fly check passes.

Gubbirk continues to have an AC of 23.
Dergaix gets a +1 bonus from being On Higher Ground (Flying).

Bite: 1d20 + 11 + 1 - 2 = 30. Hit.
1d6 + 4 + 4 = 10 damage.

Claw: 1d20 + 11 + 1 - 2 = 14. Miss.

Claw: 1d20 + 11 + 1 - 2 = 30. Hit.
1d4 + 3 + 2 = 6 damage.

52 - 16 = 34 HP remaining.

Gubbirk has this cool feat called Step Up.

Whenever an adjacent foe takes a 5-foot step away, he can make an immediate action to step after them, thereby preserving adjacency for melee.

Cool. Can he fly though?

There's nothing in the feat description that says he needs to be able to fly, it just says he can do it. Arguably he can even do it while entangled, prone, blind, and paralyzed, to pursue a hidden invisible ethereal enemy retreating directly through a solid wall.

There's no possible way I'd accept that interpretation.
Step Up does not give you wings.

Spoilsport.

He's got throwing axes as a ranged option, though it's not exactly his area of expertise.

1d20 + 11 = 28. Hit.
1d4 + 3 = 5.
Dergaix has 37 - 5 = 32 HP remaining.

Dergaix flies above his target and uses his breath weapon.

Gubbirk has +5 to Reflex saves.

1d20 + 5 = 14.
He doesn't dodge it.
He also doesn't have any protections against fire up.
Nobody told him he'd be fighting a red dragon today.

2d10 = 8.
34 - 8 = 26 HP remaining.

Throwing Axe again. He carries 10 of them.

1d20 + 11 = 13. Miss.

Breath Weapon?

A dragon can use his breath weapon every 1d4 rounds.

1d4 = 4.
Come back 3 rounds from now.

Dergaix is unsure if he is winning or not.

He is aware that he is almost half dead. In fact, if he had to put numbers on it, he'd say that on a scale of 0 to 59 he is about 27/59 dead.

The Goblin looks more than half dead, but the Goblin doesn't have to wait 3 more turns before he can attack again.

What exactly are his odds, here?

Assuming you both stay still, you breathing fire and him throwing axes, Gubbirk will on each round deal 1d4 + 3 damage on a hit, and hit on 12/20 rounds. This averages 3.3 damage per round, or about 11 rounds to the win.

Dergaix will deal 2d10 damage on a hit, or half that if Gubbirk makes the save, but he can only attempt that on average every 2.5 rounds, for an expected output of 3.3 damage per round, or around 8 rounds to the win.

Of course, this is highly dependent on the precise rolls achieved, and since you have already observed your next breath weapon recharge roll is a 4, you can move your own estimated time-to-win up to 10 rounds.

For a more exact answer, we could run a monte-carlo simulation. Starting from the current state of the fight, Dergaix ends up winning in ~41.7% of worlds. If you'd rolled a 1 for your breath weapon to charge just now, instead of a 4, then you'd be winning in ~71.9% of worlds. It varies wildly in duration between as few as 2 and as many as 18 turns before the end, but the average is around 7.5 turns.

You might also decide not to fight to the death: He can deal at most 7 damage in a round, so if you decide to fight until you have 8 or fewer HP remaining before fleeing, your win rate drops to a mere ~20.1% but you'll almost certainly escape alive (albeit very injured) if you lose.

Alternatively, we could explicitly calculate the exact answer in polynomial time by using a markov-chain that we can unravel by expressing it as a transition matrix-

You know I'm 3, right?

And if I close back into melee?

Well, his max damage output is 56, so he could one-shot you now that you're weakened, but realistically-

Pass.

Suppose I wanted to oppress these Goblins, without having to accept substantial risk to my own life, without fear of very almost dying or needing to flee with my tail between my legs, and without my whole future being determined by a dice roll?

... That's not really how this whole situation works.

Is nothing happening?
That means it's my turn, right?

Gubbirk has been thinking about this situation a fair bit.
He's not entirely certain how injured the dragon is, so far.
He knows their tribe's Alchemist can always make more, but still it represents a considerable investment of resources.

That said, he's not certain of his victory otherwise, and he doesn't want to take any huge chances here.

Gubbirk reaches into his bag and retrieves, and then drinks, a Potion of Resist Energy (Fire).

That stuff I said earlier about rounds-to-win and probabilities and stuff?

Scratch all that.
Your odds are only slightly above zero.