I plan on writing up a page on Lore-master non-combat pets complete with pics and maps, but until then you can find a good starter guide at LotRO Vault.

Confirmed non-combat pets so far: hare, sparrow, turtle, dog, cat and fox. Two have yet to be found.

You can’t have a non-combat pet and a combat pet out at the same time. And you can rename your pet by targeting it and typing /pet rename (newpetname). More info in the linked article.