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Ok, so here's the deal:
As I read it, I can make the following attacks in 1 round:
Is this correct?
That's almost correct. Your second attack from Extra Attack is part of the same Attack action as your regular attack, so it should happen before your off-hand attack.
The other thing you should know is that Action Surge gives you an extra action, not an extra attack. If you use the extra action from Action Surge to take the Attack action, you get your regular attack and the extra attack from Extra Attack.
You have to look at each piece that gives you an attack and see how they affect each other.
Extra Attack:
The extra attacks that you gain from extra attack are applied to your attack action. This is an important note because anything that gives you an extra action (or extra attack action) gives you this ability as well.
Horde Breaker:
This specifically states that it only applies once per round, contingent upon making an attack. It does not count as an action or bonus action, but as its own situational effect with specific conditions that trigger it.
Two-Weapon Fighting:
This ability takes up your bonus action (which you only get one of every round). It allows you one extra swing. It should be noted that rapiers are not considered light weapons, and thus are restricted to requiring the Dual-Wielder feat in order to use properly in this context.
Action Surge:
This one is the trickiest, as it gives you an action to use. Since it gives you an action, your extra attack ability is used with this attack as well.
Action: 2 attacks due to extra attack.
Horde Breaker: 1 attack with special targeting restrictions.
Bonus Action: 1 attack with 'off-hand' weapon.
Special
Action Surge: If you use this, you gain 2 more attacks (due to extra action).
Total:
Without action surge, you end up with 4 attacks. On rounds where you use it, you end up with 6 attacks.
Targeting isn't that complex overall, as long as you understand that the Horde Breaker attack specifically has to target a different creature than the attack that you use to trigger it. You decide which of your attacks trigger it.
This means that you can focus all but one of your attacks on a single ogre, with the Horde Breaker attack going toward the second ogre. It also means you can distribute these attacks completely evenly.
If you focus on fighter from here, you can end up with more attacks, as their Extra Attack ability upgrades over time. If you focus on Ranger from here you can get more Hunter abilities and augmentation spells that can help you out a lot. Both are viable options.
That is mostly correct. However, you cannot use Two-Weapon Fighting while dual-wielding rapiers, since they are not light melee weapons, unless you have the Dual Wielder feat. Also, you can still make one more attack during this turn, as Action Surge grants you an additional action instead of just one attack.
Here is a detailed breakdown of your turn's attack progression:
Correct. This is an attack granted by your Attack action, described as follows:
Actions in Combat - Attack
With this action you make one melee or ranged attack. -- Certain features, such as the Extra Attack feature of the fighter, allow you to make more than one attack with this action. (Player's Handbook, p. 192)
Correct, provided that the prerequisites given by the Horde Breaker feature are fulfilled:
Horde Breaker
Once on each of your turns when you make one 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. (PHB, p. 93)
So you have to make the attack with the same weapon you used in your regular attack against Ogre 1 and Ogre 2 must be within 5 feet of both you and Ogre 1.
As a side note, you could also use this feature to fire a bow at both of the ogres, provided that they were within 5 feet of each other and within the range of your bow,
Incorrect! You cannot use two-weapon fighting, because a rapier (PHB, p. 149) is not a light melee weapon. The rule for two-weapon fighting is quoted below (emphasi mine):
Melee Attacks - Two-Weapon Fighting
When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in your other hand. (PHB, p. 195)
Both the weapons you wield should be light melee weapons in order to use two-weapon fighting. So, if you were wielding, for example, scimitars in both hands instead of rapiers, this would be correct.
Alternatively, if your campaign is using Feats, you could forgo your ability score improvement the next time you level up (since Fighter 6 and Ranger 4 both grant you an ability score improvement) and take the Dual Wielder feat (PHB, p. 165) instead, since it allows you to use two-weapon fighting "even when the one-handed melee weapons you are wielding aren't light".
Correct. Your fifth Fighter level grants you the Extra Attack feature as follows:
Extra Attack
Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn. (PHB, p. 72)
To be noted here is that should you continue your level progression to Fighter 5 / Ranger 5, you can still make only two attacks per Attack action, as the two Extra Attack features do not stack.
Partially correct. Yes, you can use Action Surge here, but instead of just one additional attack, it actually grants you one additional action. Action Surge reads as follows:
Action Surge
Starting at 2nd level, you can push yourself beyond your normal limits for a moment. On your turn you can take one additional action on top of your regular action and a possible bonus action. (PHB, p. 72)
So, when you use Action Surge to take the Attack action a second time, your Extra Attack feature still grants you that second attack, so you can make one more attack during this turn.
