So I guess the "An" signifies "this coming Wednesday/Monday", but if you ask "Do you want the wedding on a Monday", would it be "ar Luan"?
Nope. ar an before the day indicates habituality (On Mondays, for example). To indicate 'this monday' you'd use Dé Luain
Because it's genitive in that case. 'dé' is the genitive of 'dia', which is an older, now rarely used word for 'day of the week'. It's a set usage.
This speaker tends to elide the n in an. This isn't unusual in ordinary speech, but it's pretty challenging in a learning environment.
The 'g' in front of Céadoin comes from being eclipsed by "ar an." Any of the prepositions (ag, ar, faoin, leis, ón, roimh, thar, tríd, and um) matched with the definite article "an," (the), means that the word following that preposition + definite article combo becomes eclipsed.
What in this sentence makes 'an gCeadaoin" and "an Luan" plural? Doesn't "an" indicate singular? Should it not be "na gCeadaoin" etc. if they're plural?
This idiom just happens to be used in plural in English. Scroll up, for "on Monday" meaning this Monday Irish would use Dé Luain, but "ar an Luan" is the habitual form.
I really don't understand how this can be plural. Why aren't multiple Mondays na?
The Irish idiom translates as an English plural because the Irish idiom refers to habitual actions, despite the day being singular. Some English dialects also use a singular weekday for a habitual action, e.g. “I like to take a walk of a Sunday” = “I like to take a walk on Sundays”.