[RPG] How many attacks does a dual-wielding Ranger get with Horde Breaker and Extra Attack

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.

So, first up, we need to make something clear: The (capital A) Attack action is a specific action defined on page 192 of the PHB. It lets you make a (lowercase a) attack, or multiple (lowercase a) attacks if you have the Extra Attack class feature.

Volley is a specific action that lets you make as many (lowercase a) attacks as there are creatures within 10 feet of your chosen point. So the Volley action and the Attack action are entirely separate, distinct actions. Since Extra Attack only works with the Attack action, there is no interaction whatsoever between Extra Attack and Volley. You can use your action to take the Attack action, or you can use your action to Volley. Extra Attack will not affect Volley in any way.

Next up, Horde Breaker and Volley. You can use Horde Breaker whenever you make a weapon attack. Since Volley involves making weapon attcks, yes, Volley can trigger Horde Breaker. I'm not sure what you mean by "extending the range" of Volley - Volley allows you to attack all targets within 10 feet of a chosen point. Attacking an additional target with Horde Breaker won't change the target or range of the Volley, it just lets you make one more attack.

Finally, Foe Slayer. This lets you add your Wisdom modifier to the attack or damage roll of one (lowercase a) attack per turn. So you can use it on the Horde Breaker attack, you can use it on one of the Volley attacks, you can use it on one of the attacks you make with the Attack action - but only ever on one attack.