Thylacoleo can only climb up the gateway part (e.g. The stone part of stone behemoth gateway) not the actual gate istself( the opening and closing part). This goes for all other gates to. Also if you didn't know already, you jump and hold space to climb.
Thylacoleo are capable of pinning down flyers in mid-flight (anything but the Quetz and Wyvern, I believe), other Thylacoleo, and even Therizinos. You can also jump from object to object by holding down the jump button while pouncing from a tree or cliff (have to hit it at the right angle, however). Upvote to inform others!