Interesting question about the next potential Avatar: As the Avatar has had animal companions (Roku and Fang, Aang and Appa, Korra and Naga, etc.); what animal companion could the next Avatar have? It would likely come from the Earth Kingdom. Fang, Appa and Naga were all ridable so, maybe it's something large enough to ride?
Could be anything, really. Badgermoles are the original earthbenders, but they're pretty unwieldy, so while Air Avatars have their sky bison and Fire Avatars have dragons, it seems to be different for each Water and Earth Avatar (it is my headcanon that Pengpeng was Kyoshi's true animal companion in time)
In my opinion I would love to ride an eel hound, though I’m pretty sure it would be terrifying, I’d probably do it anyway. As for long distances, we’ve already seen that in book 3 of A:TLA, so we know they can go very far very quickly. Also, from what I can remember, they seemed to be quite gentle so they could form quite a strong bond with the Avatar. Also, they are animals native to the earth kingdom and the next Avatar is earth kingdom so the Avatar could meet one very young, much like how Korra found Naga.