What exactly is なり?
It is a conjugable suffix (助動詞). It attaches to the attributive (連体形), a substantive, or an uninflected adjective. It expresses designation (指定) or predication (断定). It is basically equivalent to である or だ. You may consider it a copula.
As a conjugable suffix, it has multiple forms: nar-a, nar-i / ni, nar-i, nar-u, nar-e, nar-e. Etymologically, it is a contraction of particle ni + verb ar-.
Is it なる?
As indicated above, nar-u is the attributive form of the suffix nar-. However, you are probably asking if it is the verb 成る, and the answer is no; it is a suffix as explained above.
Is it 也, an archaic version of です or ある?
If so, does that mean that in the past, Japanese verbs did not all end in the う-sound?
If by end in -u you are referring to the basic conclusive form (終止形), then yes. The verb ar- in conclusive form is ar-u (ある) is modern Japanese, but it was ar-i (あり) historically while the attributive form (連体形) was ar-u (ある). And as indicated above, this nar- is a contraction of particle ni + verb ar- so has a conclusive form of nar-i.
Note that there is also a hearsay or inference nar-i as well. This is a different word and attaches to the conclusive form rather than the attributive. Do not confuse these two. Here is a well-known example using both nar-.
"A woman, too, will try her hand at the diaries that men are said to write."
(From Tosa Nikki, c. 935)
Notice that the first nar-u attaches to su, which is the conclusive form of the verb, so it is a hearsay nar-. And the nar-i at the end of the sentence attaches to suru, which is the attributive form of the same verb su, so this is the predicative copula.