I'm getting tripped up by A/The quite a bit...it seems arbitrary.

This grammar is pretty tricky when you first learn it. You could think of it as "the bag having baby" but that's not good for long-term understanding. The more you practice it, the more it'll click. It's almost like how adjectives are put before nouns. Spicy chicken = 매운 닭고기. You use the 는/은 or -ㄴ and then the noun. Now if you describe more...The chicken with no head = 머리가 없는 닭이.

More accurately, think of 가방 있는 as a compound adjective modifying 아이, so it would be, the bag having boy, or to make it less confusing, the boy having a bag, or the boy who has a bag, etc. etc.

Because in English, when you have a countable noun that is singular in number, like "bag", it has to be preceded by an article, either "the" if it's a specific bag you've been talking about, or "a/an" if it's any one bag.