"Dè an t-ainm a th' oirbh, athair?"
Translation:What is your name, father?
At first glance, this doesn't strike me as a very natural question. Or does the "athair" refer to "Father" as the priest? In that case, "Father" should begin with a capital "F" rather than with a lower case "f". (Nit-picking, I know, but Duolingo rather pushes learners in that direction...)
Yes to both.
If memory serves ainm is feminine and since it starts with a vowel it gets an t- as its definite article.
Th' is short for tha. In fact most instances where you see an apostrophe in Gaelic are because the apostrophe ks replacing a vowel. 'S for example is short for is.
Tha approximately means "is".
The full phrase literally means "what is the name that is upon you? Father."