> "wegen den Regen" means something slightly different (more like: "because of _multiple_ rainfalls")

That's your natural feel of language, and you are deriving from casual use of Dativ plural ... but in these situations, Genitiv would be correct again (wegen DER Regen, but more clearly: wegen der Regenfälle, as Regen is uncountable (unlike, for example, Sturm/Stürme)).

Your example is vernacular German as spoken on the road, but grammatically, it is incorrect.

Yes, I am lots of fun at parties.