I'll take a stab at this.
"Ansin" can mean "there" indicating a place that contrasts with "anseo" meaning "here." As you point out, "ansin" can also mean "then."
"Ann" is a conjugated preposition meaning "in it." It can also mean "there," but not quite the same "there." The difference between "ann" and "ansin" when they both mean "there" is something like the difference between the two occurrences of "there" in "There are cats there." "Ann" is more like the first "there."
But Irish works differently than English. "Tá cat ansin" means "There's a cat there." You don't need a word for each of the English "there" words. On the other hand "Tá cat ann" means "There's a cat." I don't think that "Tá cat" without "ann" would be a full Irish sentence. It would be like "is a cat" in English.
It's important to notice that "There's a cat" and "Tá cat ann" both have more than one meaning. One meaning is just about existence, not really location, as in: "Who are the characters in the story? --Well, there's a cat." Another meaning is more about location. After looking for a while for a cat, any cat, you might say with satisfaction when at last you find one: "Ah, there's a cat." You have to emphasize "there" to get that meaning. Or in answer to the question: "Cad atá sa seomra?" you might say "Tá cat ann." I think you can justifiably translate that sentence as "There's a cat in it", "There's a cat" or even "There's a cat there."
"there" is ALWAYS an adverb--not a pronoun. ever. not even possible. My English teacher would probably have a heart attack if I called it one. a pronoun must RENAME another noun, or just perform the same function as a normal noun--in other words, it must be a person, place, or thing. "there" is most certainly not any of those. "more" is an indefinite pronoun, and "there" is an adverb describing where it is. But people tend to be weird and add an extra one in the front of the sentence just because. Frankly, I think it often sounds better, but it really isn't necessary, or good grammar.
I'm not going to debate the merits of your teacher, but given a choice between your teacher and the dictionary, I'll take the dictionary.
pronoun 7 (used to introduce a sentence or clause in which the verb comes before its subject or has no complement):
There is no hope.
pronoun ˈt͟her , sense 1 is also t͟hər Definition of there
1 —used as a function word to introduce a sentence or clause
there shall come a time
2 —used as an indefinite substitute for a name
1 pronoun - impersonal subject
there's a man outside tá fear taobh amuigh