Extra Attack only modifies a regular Attack action

Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.

I think the fact that "Attack" is capitalized seems to implicate the Attack action under actions in combat vs. any attack.

Horde Breaker could be used in conjunction with Whirlwind Attack

Once on each of your turns when you make a weapon attack, you can make another attack with the same weapon against a different creature that is within 5 feet of the original target and within range of your weapon.

Since Horde Breaker triggers off a "weapon attack" not an "Attack action" you should be able to use Horde Breaker to target one of the enemies next to you. Its simply triggering off one of the attacks made as part of the whirlwind.

Optimization breakdown

Since all Hunters get either Volley or Whirlwind, the Extra Attack class feature is there as a stopgap in the meantime and also a fallback for when there is only one enemy in range to be targeted. Horde Breaker should still be viable even though its utility will diminish in comparison upon reaching level 11 as a ranger, but even then its still one more attack per turn.

Yes

There is nothing that prohibits this interaction. The ranged attacks made from Volley count as weapon attacks, so any of them can be used as the "trigger" for Horde Breaker.

Note however that Horde Breaker allows you to do this only once on each of your turns, so you can't trigger it on each of the Volley attacks, but only on one.

In your example, you can attack each of C1, C2, and C3 once from Volley, and another one of them a second time, as long as the target of the Horde Breaker attack is within 5 feet of any of the original targets (so if C3 is further away than 5 ft from both C1 and C2, C3 will not be eligible for the Horde Breaker extra attack).