So does a neuter noun not change its ending in the accusative ever, or are there exceptions?
Neuter nouns' accusative is the same as their nominative since Ancient Greek.
Could you use "γυαλί" (instead of "ποτήρι")?
If you are too cool for school ;) It's slangy.
That's true. When you say "γυαλί" you refer to the actual material rather than the object. xD
Why not τον ποτήρι?
Ποτήρι is neutral and it takes το in accusative case.
couldn't φέρνει mean carry, too?
He carries the glass. = Μεταφέρει το ποτήρι.
I think that "carry" doesn't convey any direction, contrary to "bring". :)