If I played Bellows Breath on a Stellacorn and then after that resolves I play a Thousand Tail, will the Stellacorn die?

Yes, the Stellacorn dies.

Here's why: damage is "marked" on a unit and doesn't reduce its Might. Bellows Breath leaves 1 damage on the Stellacorn Herder (3 Might — not enough to kill it). Then Thousand-Tailed Watcher's "when you play me" trigger resolves, giving enemy units −3 Might this turn (minimum 1), dropping the Herder to 1 Might. Since the 1 marked damage now equals its Might, it's killed in the following state check.

If you paid the [Repeat] cost on Bellows Breath, the Herder would have 2 damage marked and die just the same. Either way, the −3 Might is what makes the earlier damage lethal.