The rules in regards to mounted combat are a little muddy.
For Reference, here is Overrun
As a standard action, taken during your move or as part of a charge,
you can attempt to overrun your target, moving through its square.
From the Mounted Combat Rules:
Your mount acts on your initiative count as you direct it. You move at
its speed, but the mount uses its action to move.
This section implies that it would be the mount attempting the overrun maneuver in order to move through an opponent's square.
Several of the feats that you are referencing imply that you are the one doing the overrun. For example, take Trample
While mounted, you can ride down opponents and trample them under your
Benefit: When you attempt to overrun an opponent while mounted, your
target may not choose to avoid you. Your mount may make one hoof
attack against any target you knock down, gaining the standard +4
bonus on attack rolls against prone targets.
Here's how I would rule this. I would say that you and only you need to have Improved Overrun/Trample/etc, however you would have to spend your own standard action in order to attempt the maneuver in exchange.
Taking a look back at the 3.5e SRD, we have this gem under the Overrun maneuver
Mounted Overrun (Trample)
If you attempt an overrun while mounted,
your mount makes the Strength check to determine the success or
failure of the overrun attack (and applies its size modifier, rather
than yours). If you have the Trample feat and attempt an overrun while
mounted, your target may not choose to avoid you, and if you knock
your opponent prone with the overrun, your mount may make one hoof
attack against your opponent.
Note that this doesn't say that "The mount makes the overrun attempt", it says "you" make the overrun attempt and then use the horse's strength to resolve it.
Since there is a lack of this clause in Pathfinder, and it's bloody confusing, I'm going to have to go with this is what they intended.
Now this doesn't stop you from having the horse make an overrun attempt on its own while you do something else, but then your horse would need the aforementioned Overrun feats, and wouldn't benefit from you having Trample (Or even it having trample feat, since it's you know, not mounted... it would need the Trample (Ex) special attack for that). Also, it would probably need to be taught an "Overrun" trick, since I personally don't believe that "move" covers it.
So I further affirm that these things lead me to believe that Rules as Intended is for you to make the overrun attempt if you want to benefit from the trample feat, and if you are the one doing the overrun then you get to benefit from Improved Overrun.
Plus, teaching an Int 2 horse additional feats is a pain.
Your friend is basically wrong.
The mounted combat rules are not very well written. But the crux of the matter is that there is a distinction between you charging while mounted, and your mount charging while you are mounted.
From the Mounted Combat section of the SRD:
Your mount acts on your initiative count as you direct it. You move at its speed, but the mount uses its action to move.
Mechanically, the mount spends its action to grant you its movement, rather than moving directly itself.
In other words, when you "move" while mounted, the mount spends actions. When you charge while mounted, the mount spends actions, but you are still the one who is charging.
Particularly bad is this paragraph:
If your mount charges, you also take the AC penalty associated with a charge. If you make an attack at the end of the charge, you receive the bonus gained from the charge. When charging on horseback, you deal double damage with a lance (see Charge).
You really have two different clauses here. The first is what happens when your mount charges (instead of granting you movement):
If your mount charges, you also take the AC penalty associated with a charge. If you make an attack at the end of the charge, you receive the bonus gained from the charge.
The second is the special rule for lances, which triggers "when you make a charge while mounted:"
When charging on horseback, you deal double damage with a lance (see Charge).
When trying to run at someone and hit them with a lance, the mechanics are:
You take the charge full-round action.
The mount spends its actions to grant you its movement.
You stop at the edge of your reach, and poke it with your lance.
By contrast, I suppose you could command your mount to charge.
The mount takes the full-round charge action.
The mount moves to within its reach of the target.
The mount attacks.
You may attack, if able (but generally not with a lance, because your mount is too close to the target).
By RAW, No.
By RAW Charge Through does what it says it does - lets you Overrun (not Trample) one foe.
It is, however, a very reasonable houserule to allow an animal companion with Charge Through (the Charger archetype gives them the ability to choose it as a feat, useless on a rider as it says 'you' not 'your mount') and the Trample special ability to combine the two and automatically 'Trample' any enemies of the appropriate smaller size between it and it's Charge target.
This is, however, a House Rule, and by RAW does not work.