This isn't an issue with the skills/rules, but, rather, an issue with your DM, who is allowing creative use of skills to have tremendously powerful effects.
The monster knowledge checks are entirely standard, though it's worth mentioning that you (generally) only get one attempt per character against a given monster type per combat, and only learn the effects of their powers if you beat the hard DC, which an untrained character with low wisdom should almost never do.
The Nature skill does indeed contain rules for Calming Beasts (phb 186). It's a hard DC of the beast's level that requires a standard action and automatically fails if you're in combat with the target, and on a success "the beast is calmed," which I'm putting in quotes because there are no rules whatsoever that describe what that means. To me it seems meant for convincing a snarling wolf not to attack you, rather than a near-magical instant way to gain an animal companion, but this is intentionally left open to interpretation.
It also contains rules for Train Animal, but that block both specifically calls out that that's mostly for skill challenge purposes, and seems to require that you already have a pre-existing relationship with said animal.
Interacting with an animal beyond those two items isn't covered in the Nature rules, but the Hunter/scout class's Beast Empathy Wilderness Knack grants the player a +2 bonus to the normal social skills (bluff, diplomacy, intimidate) against beasts, which implies that any true "social" interaction uses those skills, even against animals. It also specifically states that the animal is under no compulsion to obey you, even on a success (hotfk 149) .
The intimidate rules similarly don't handle the use case your DM permitted. In fact, the standard "intimidate" action out of combat specifies that Whether or not the check succeeds, using this skill against a monster usually makes it unfriendly or hostile toward the adventurer. (phb 186), making it an unlikely-intended route for obtaining a mount.
He's not wrong to do this - he's just outside of the scope of the written rules, and once you're beyond the scope of the written rules, what works for your group should be dictated by what works for your group.
If you don't enjoy this, have a friendly chat with your DM, but realize that you'll probably lose your cubs as well.
One of the most popular DM tropes is to always respond to your players' ideas/requests with "yes, and," meaning that you should (generally) allow nearly any given course of action, but provide twists. If your DM is familiar with this concept, he may just be being a tad too permissive (though again, this is all a matter of style, with no one correct answer). If your DM is unfamiliar with it, he may be feeling pressure to not shut down ideas presented by the players.
What "yes, and" means, really, is that a DM should let players dictate what actions they take, but should not let players dictate what results follow those actions. You want to intimidate the shopkeep into giving you a discount? Okay. He cowers and opens the cash box, turning it towards you. With your passive perception you see a faintly glowing line of text on the inside lid. Looking closer, the inscription reads, "I prepared explosive rune this morning."