Do abilities have a priority ranking?

Skills that prevent/guarantee doubles cancel each other out, and thus will use the default to determine if a unit doubles, check to see if one units Speed stat is 5 higher than the other.
In your example, Armads allows Hector to double, but axebreaker cancels out the followup.
Niles is able to double regardless of his speed since axebreaker allows a follow-up on axe users.

There are many skills that prevent or 'guarantee' doubles, here are a few:
- Prevent Doubles: wary fighter, the breaker skills and wind/watersweep (but only yourself).
- Cause Doubles: breakers, bold/vengeful fighter, brash assault and quick riposte.

Skills in opposing categories will cancel each other out. You can also have multiple of either category but only one skill will cancel one other skill. So if you have Armads, wary fighter in the B slot and quick riposte in the seal slot, Armads and wary fighter cancel each other out, but quick riposte will still allow a follow up.