You're trying to use a D&D precedent in Pathfinder, but the Diplomacy rules were specifically changed!
The line "You cannot use Diplomacy against a creature that does not understand you or has an Intelligence of 3 or less" was explicitly added to the skill description in Pathfinder; it was not present in prior incarnations of the rules. Given this, it hardly matters whether it worked in 3.5 or not.
what is this witch ability good for if it doesn't allow for using diplomacy?
Um, talking to animals, same as the spell it emulates? You can still get information from them. I believe you could also cast [language dependent] spells on them, if you want a more specific game benefit.
Roles Can Be A Bummer
Making actions/options not available to people can force people out of the magic circle pretty quickly. Sure, roles can help people feel special, but it can go overboard. This does not mean that the rogue shouldn't be stealthy, or that the cleric can't be tank-y, but not being able to use or do something because of an arbitrary class restriction breaks the circle, which means less fun for everyone.
Sometimes, multiple people actually can do the same thing, and that should happen sometimes. If every task you throw at the player characters cannot be accomplished except for 1 person's specialization, it may be a harder adventure than these players (as a group) are fit for. Not that there can't be a situation where you need a particular player to do a something, but that it shouldn't happen all the time.
Perhaps It is a Matter of Perception
So maybe this player is simply just does not imagine the same thing you do. You can try to reinforce what you envision by describing what is going on. Is she/he making informed choices? Does she/he realize what the gravity of the situation, or how her/his character has certain strengths?
Does this player realize the cleric is running around in thick armor, while that player's character is running around in normal clothes? For example, if I think the cleric is running around in normal clothes, and goes into combat and comes out unscathed, I could run in my normal clothes, and also come out unscathed.
If this is the problem, you simply need to be more descriptive! Describe how the cleric is wearing armor, or how the rogue is especially cat-like, and how this player's character is not.
For Games Which Depend On Roles
It may be time to have your other players step up. Have the cleric yell at the other character to stay back, or that the cleric "has this." Alternatively, the cleric may ask "PEASE SUMMON THIS TO HELP ME!" as he charges in.
If the rogue attempts to sneak around, the rogue can give specific instructions; "Can you make a distraction over there?" "Wait here, and if I'm not back in 15 minutes, go get the others to rescue me." Or even: "this looks super dangerous. I don't think you'll make it. We don't want to get ourselves killed, yeah?"
In short, give him tasks, or have the other players give him tasks to help with things. Talking is a free action in combat; use it. This is especially good for players who may be experiencing some mental handicap; it gives them something concrete to act on. It allows them to contribute (which feels great and is fun) without going through negative experiences.
Finally, a player or the DM can specifically highlight when certain actions will require a specific specialization, and who has that training. If it becomes obvious that a task is dangerous and requires training, then most people leave it to the person with the training. The DM can further forbid people without specialization from trying, stating that it's obviously too hard. This is more "hand-holdy" than some people like it, but sometimes people just need their hands held.
Talk About It
Talk with the player about the roles. What is her/his character good at? What should she/he focus on? These other characters have something special about them, what is she/he special at?
Talk about how their character is so good at magic/whatever else, and how they should try to make their magic/whatever else be the solution for the current situation. (After all, we're solving these situations using our strengths, use your strengths to help solve it!)
Finally, you may have to resort to letting the consequences of this player's action happen. Let the events, despite other character's best efforts, happen. Talk about how that character was not focusing on what they were good at, or how their play-style does not match up well with the class they chose. Make a new character that does okay at everything, such as most games' version of a "bard." Note: this does not mean make a character who is rule-breaking and superior to all the others, but one that stands a decent chance at performing many things, and isn't a large risk to the success of most tasks.
Since the general rule is that everyone can make Attacks of Opportunity, so can Animals unless explicitly forbidden.
There is nothing in the rules, at any sane location, that prevents animals from taking Attacks of Opportunity, such as:
Intelligence has absolutely no bearing on Attacks of Opportunity. In fact, going one step further, even Mindless creatures like Oozes or Vermin can take Attacks of Opportunity.
About the Flank trick
Your friend might be thinking of the description of the Flank trick from Handle Animal, which includes
(emphasis mine). In the context of this trick, this sentence makes little sense to me. Taking AoOs has little to do with flanking. Also, by RAW, this forces the animal to take AoOs, where without the trick it had a choice.
I find it reasonable to assume that RAI is that the animal always provokes attacks of opportunity if needed to get into flank, which also makes the context actually make sense.