This has come up before.
One argument is that you cannot raise an animal's Int over 2... because it stops being an Animal then and is no longer suitable as an Animal Companion.
On the other hand, some argue that it is a class feature, and the player should be able to do that.
For Organized Play, the PFS FAQ has an answer: yes, you can raise the animal's Int over 2, but all you get is 3 more tricks. It does not become able to speak, cast spells, plan, or any other effect of being sentient.
There are also a couple of older blog posts around here somewhere that address the issue in more detail.
I play it as no. I don't fee like arguing whether gorillas can gain proficiency with a greatsword.
Yes, you can. However, it's limited to a more intelligent animal intelligence.
You still need Ride and Handle Animal checks.
They can understand a language. So you can say, "Attack the goblin with the red shirt," make the Handle Animal check, and it attacks the goblin with the red shirt instead of the nearest goblin.
So everything that has been said so far is great for druid style ACs, but paladin mounts start with an int of 6, well over the animal range. So Ride checks to stay in the saddle or manage physical tasks like fighting from the saddle or quick mounting make sense, but does it still use tricks? Handle animal to push it? Give it one point of additional int and it matches some pcs afterall.
The ride check is for more than just communicating with the animal. It also covers things similar to balance while riding the animal.
Handle animal is pretty questionable for a paladin mount, because of their magically innate intelligence. According to guidelines in Ultimate Campaign, you don't need handle animal to get your mount to do what you want. But those rules are not included in PFS.