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)
- Horde Breaker attack on Ogre 2
Correct, provided that the prerequisites given by the Horde Breaker feature are fulfilled:
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,
- Off-hand attack as bonus action for fighting with 2 weapons
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".
- Extra fighter attack from lvl 5 fighter
Correct. Your fifth Fighter level grants you the Extra Attack feature as follows:
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.
- Action Surge attack from lvl 2 fighter
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:
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.