On this, among others, I can't hear her say "an". I hear "Tá uisce ag a gcósta". This is fine, I can live with it. But as it happens in a few places:
Is it a dialectical thing? Or just a minor error?
Pretty sure that's intentional, though I'm not sure which specific dialects it belongs to.
"At the coast" does not sound idiomatic to me. I would always say "on the coast."
"I collected shells at the seaside", "we had a picnic at the shore" and "there's a nice breeze for flying kites at the coast" are all perfectly normal, idiomatic uses of "at" referring to the margin between the land and the sea.
Imagine yourself crossing a desert, heading towards the coast, because "There is water at the coast".
does "ag" mean both "has" and "there is", only being dependent on the context?
It's not ag itself that has the meaning "have". It's ag when used with a form of bí. For example, Itheann muid ag an gcearnóg could never be interpreted with "have".
However, when used with bí, yes.
Not that it makes sense, but could this sentence mean "The coast wants water" as well?
Why is it that 'go dtí an cósta' has no séimhiú or change but 'ag an gcóstá' has an urú change?
Again... Water is at the coast, how do we know when it means "there"? I deduced that's what they wanted. Is it just the English translation
It's not just the English translation, it's just the preferred way to say it in English.
Take the sentence Tá úll ar an mbord. You can translate that as either "An apple is on the table" or "There is an apple on the table". Most native English speakers would say "There is an apple on the table". If you add a definite article - Tá an t-úll ar an mbord, you can really only translate it as "The apple is on the table".
So when you have an indefinite article (or no article at all in Irish), you have a choice of which form to use in English, but most of the time "there is ...." is the preferred structure in English. So "Water is at the coast" isn't technically wrong, it just sounds weird, and if you reached that translation by doing a literal, word for word translation of the Irish sentence, I think it should be considered incorrect, because "There is water at the coast" is a much more natural translation.