If an 8 might attacks my 9 might holding can he then instantly bellows breathe and kill my unit?

Yes — but only if he plays Bellows Breath during the Showdown, before combat damage. Bellows Breath is an Action with "play on your turn or in showdowns," and the attacker has Focus first, so he can cast it while the Showdown is Open.

The 1 damage stays marked, so your 9-Might unit only needs 8 more from combat to reach lethal. During Combat Cleanup, lethal units are killed (Rule 323.5) before the heal is inserted (Rule 466.1.a.1), so it dies — though his 8-Might unit also takes 9 from your unit and dies.

If he waits until after combat resolves, no — Combat Cleanup heals all damage, so 1 damage alone can't kill a 9-Might unit. (Assumes he assigns all 8 combat damage to your unit; extra defenders would change the math.)