Towering Mountain of Rage
Right, you want to to propagate Rage effects throughout a tower of creatures. That's... actually kind of interesting. Beast Totem chain grants Pounce and 2 Claw attacks, substantially improving the combat prospects of all creatures in the tower.
Well, the applicable rules are contained here, Pathfinder's Mounted Combat Rules on PFSRD.
Looking them over, we have some... interesting interactions. For a start, as your Half-Orc mount is not a 'combat-trained mount', you have to make a move action to control him in battle as he becomes frightened. Yes, the condition. No, I am not making this up.
But hopefully we could waive that somewhat ill-considered clause (what about an intelligent animal? ugh) and assume a half-orc with player class levels is 'trained for combat riding'. So, you're looking at a DC 5 check, probably with a -5 since a half-orc is not 'suited to riding', to keep the use of both your hands and 'control him with your knees'.
I'm struggling to contain my giggles as I write this.
A slightly more difficult attack is to 'Fight With A Combat Trained Mount', a DC 10 Ride Check, with again that pesky -5 for being ill-suited as a Mount. With Ride as a Class Skill, though, it's probably pretty easy to make this check, especially with an Exotic Military Saddle.
Note that EACH rider would have to make this check, so 3/4 of the stack if you're going Mammoth->Gorilla->Half-Orc->Gnome. I'm not sure if Gorillas get Ride as a class skill. This is particularly problematic as if the Gorilla gets unseated from the Mammoth... well, actually, both the Half-Orc and the Gnome would get to roll to 'Stay in saddle', actually, which is only a DC 5 check. Presumably the Gorilla could attempt to 'Fast Mount' in his next turn, thus returning the stack bonuses to the Mammoth.
I can't find any rules in the Mounted Combat section or the Ride skill to disallow this.
I also think it would be awesome, and utterly approve of the mobile 'Tower of Rage'. Note that the Entire Stack would act on the Initiative of the gnome ("Your mount acts on your initiative count as you direct it. You move at its speed, but the mount uses its action to move.")
I assume this means that the stack acts on the gnome's initiative, but moves at the Mammoth's speed (and if it charges, they all count as charging). Note, I am unsure if a Small Character with 10' of reach from a spear could actually reach the ground if he was on the back of a half-orc on the back of a gorilla on the back of mammoth. I think he couldn't, actually. Ergo, he should probably multiclass to bard and do some form of 'rage music', because, why not.
There is a catch, however; Ferocious Mount mentions that you must spend an extra round of rage to spread the rage to your mount, which the mounts can't do as they have no rage class ability. Greater Ferocious Mount makes no mention of this, but Rage powers don't activate if the wielder is not in a Rage, SO while you can make a Tower of Mounts, you cannot make all those mounts rage unless each mount in the stack is also a Barbarian.
It's worth noting that if each mount is a Barbarian, they will pass on any continuous Rage powers during the Rage to any Barbarians lower in the stack, potentially making the lowest Barbarian have many many many Rage Powers.
But yes. Unfortunately as is it doesn't work unless all but the lowest 'mount' in the stack are all Barbarians with enough Rage rounds to make this worth doing.
Awakened Cat Barbarian riding a Gnome Barbarian riding an Orc Barbarian/Warchanter with war Drums riding a Minotaur Barbarian riding a Huge-Ass War-Mammoth is essentially 90% of the point of playing Dungeons and Dragons.
In 3.5e they would all be wielding monks, as monks are a manufactured weapon.
Probably it would end up looking a little bit like this;
YOUR LIGHTNING IS ALL I NEED
From a strict reading, the mount does not have Share Spells (even though it's an animal companion, and they usually do) because it's a Beast Rider mount (different from normal Animal Companions). Note here that it does not say that it loses the Share Spells ability, merely that it does not gain it. However, it is also a Cavalier mount, and therefore should be treated as having Share Spells for the purpose of qualifying for archetypes. (This is a specific exception, as Cavalier mounts usually do not have that ability. This mount also does not have that ability.) There does not appear to be a conflict here.
By extension, RAW it should be able to qualify for animal archetypes (and only animal archetypes) as if it had Share Spells.
Technically, if you really wanted to cheese it for RAW, you could argue that it would continue to qualify as such even if you had taken another archetype that also removed Share Spells, but that, I think, would be pushing things.
Luckily, both classes are on http://www.d20pfsrd.com/
So in your specific example, your Cavalier looses his wolf, and gets it replaced with a mount of his chose, for example really really similar wolf. This new wolf would be treated as animal companion of a druid with effective druid level of mammoth rider's class levels + cavalier levels because you use cavalier levels as druid levels for this purpose, and mammoth rider's levels stack. So, by the rules, it is different wolf with almost the same stats and bonuses...
Sadly, because by the RAW it is not the same mount, Cavalier's special bonuses like free armor proficiency for mount, or lack of armor check penalty, would no longer apply. If I were DM, I would consider house ruling them to work, but that's not what rules says.