It's got to be a coincidence that the word for "years old" and the word for "son" are the same, right? Or is this supposed to more literally mean "I am a son of eighteen"?
It's not a coincidence, it is the same words, it has an additional meaning. I think it is used in this idiom in a similar way to the way it's used in expressions like בן כפר, בן קיבוץ, בן ברית.
Aní ben shmoné esré.
Do numbers not work? I wrote 18 instead of שמונה עשרה
In learning terms, that's cheating
Is the female form of eighteen used irrespective of whether it is a man or a woman speaking? I know numbers take the female form by default but here it is a man speaking.
Well, yes, שְׁמוֹנֶה עֶשְׂרֵה is feminine, because you do not speak about eighteen sons, but about a son of eighteen.
Almog, what do those three idioms mean?
Well, בֶּן־כְּפָר "son of a village" should be a villager, בֶּן־קִבּוּץ I suppose is the same as a קִבּוּצְנִיק and בֶּן־בְּרִית "son of the covenant" is a jewish fellow believer.