I am assuming that the questions you pose aren't rhetoric, but actually questions! If I am wrong in this, feel free to ignore or delete my response.

Other rivers that are described as having "mouths" are the Siril (in Hyarrostar, Númenor), Baranduin, Entwash and Anduin. The idea is that the river doesn't flow into the next bigger body of water in one singular channel (a single mouth) like, for instance, the Thames, but in several spread-out distributaries like the Danube or Nile - in modern English we would use the Greek word "delta" to describe it, but perhaps Tolkien felt that "delta" would have sounded too technical.

Tolkien himself translated Arvernien as "land beside the Verna" (Parma Eldalamberon 17), which suggests that "ar" isn't the royal title (derived from "aran") but rather the prefix "ar" denoting "near, next to, beside" (derived from "ara").

As for Verna, nobody knows anything for certain. Strictly speaking Vairë should turn into Bêr- (we see this consonantal shift from /v/ to /b/ in Q Valar = S Belair, Q Varda = S (El-)bereth, Q Voronwë = S Bronweg, etc.), but the prefix Ar- may trigger mutation from /b/ to /v/. Not sure where the /n/ should come from, though; Arvêrien would be a more likely name if Vairë were somehow involved.

Other theories assume that Verna is simply a place name related either to the root WER "winding" (suggesting the riverbed of Sirion) or PHER "beech" (perhaps as an alternative name for the beech-forests of Nimbrethil).

I've also seen suggested (by David Salo I think) that Arvernien may originally have been coined to recall the French Auvergne (just as Beleriand was inspired by Broceliande), but of course that needn't mean Tolkien didn't put additional thought into the etymology! :D

OK, shutting up now! Thanks for the food for thought.