Looks like Yamato has a dog-related fruit.
Now, concerning Zoro going to King or Kaido.
It can go either way.
On one hand, Oda could very well have the Momotaro reference and have Luffy (monkey), Marco (bird), and Yamato (dog) temporarily fight Kaido altogether. The problem with this is that:
If we follow the popular assumptions, then Marco (who is apparently very low on stamina) shouldn't go to Kaido (someone much stronger than King).
Or on the other hand, Zoro goes back to Kaido. Probably at the same time as Luffy. Hence why their recoveries are paralleled.