Yes, Wapiti is also Algonquian--the basic meaning is "white" or "pale"; I suspect that it refers to the animal's rump, which aligns it in meaning with the name of the European bird 'Wheatear' (as it stands, 'wheatear' is pretty meaningless, but it's another case of avoiding embarrassment--it was originally "white-arse"). If I understand correctly, there was an effort made to make "Wapiti" the standard term in American English to avoid the confusion caused by the different meanings of 'Elk'; it is also nicely balanced in that both Alces alces and Cervus elaphus would thus have Algonquian-derived names. But of course the elk hunters would have none of it.
You can see that "Alces" and "Elk" are cognates; that suggests that possibly we have an Indo-European root here, but I don't have the resources at home to check this and of course on Memorial day the libraries are closed.
I have eaten elk = Cervus elaphus (very tasty) but never elk = Alces alces.
Btw about the lagomorphs: Slavic _zaiats_ is often identifies as 'rabbit' (same for Latvian _zak,is_). This is not right; it is 'hare'; only the hare is native to eastern Europe. The rabbit (Slavic _krolik_, Latvian _trusis_) is known only as a domestic animal. The rabbit was probably introduced to the British Isles by the Romans, which would explain why the name ('coney') is of Latin origin (_cuniculus_). "Rabbit" is from Walloon French, which got it from Flemish, and originally designated only the young.
The embarrassment felt by some at pronouncing the animal's name "cunny" was dealt with in several ways: 1. by redifining "rabbit" to mean adults; 2: by changing the pronunciation of "coney" to make it rhyme with "phony balony"; 3: by changing the first letter to 'b' and coming up with "bunny," suggested by the use of 'bun' for the critter's tail, so called because of its shape (cf. the bun produced by a bakery, or the bun that keeps a woman's hair off the back of her neck). I was once present when the person reading Ps. 103 said: "so are the rocks for the connies." Since there was a Connie in the group, there was a deal of not quite successfully suppressed laughter.
"Rooster" was apparently formed from the term "roost-cock" to avoid the sexually suggestive word, but it remains an Americanism. Brits are less inclined to think of erotic anatomy when the encounter the word "cock," or at most will say "cockeral" or "cockerel" (I find both spellings).
Stephen