Yeah no. You have mounts only obtained through leveling certain races and classes. One that is a super rare drop from Timewalking. You also have one that is only obtainable with a 2014 promotion.
There was a bug on that that was fixed last year for eiligible accounts that finally granted it to a few who were missing it.
IDK. Good luck sorting out the mog options. There are many and Wowhead has great tools and a dressing room to try items on.
As for options, Blizzard has been adding tons - including the types you asked for - via the Trading Post. Old mogs offered do return a they cycle through.