Here’s the breakdown for anyone visiting this thread:
Battle Bird and Penguin are form the TP shop, they cost $25 USD each. They can be mounted by any class.
Red Panda is a reward for turning in 20,000 PvP points. Not sure about riding status, probably only by select classes.
Guinea Pig is a reward from a specific Guild mission, which rewards the egg into guild storage and the guild leaver sends it to a guild member. The Guinea Pig is only mountable by sleect classes in iTOS, but on kTOS it’s mountable by any class.