Because that’s the Gaelic way to talk how you feel – you use definite nouns for that. I think it is possible (and at least some speakers might speak this way) to not use a definite article (so chan eil pathadh orm instead of chan eil am pathadh orm) similarly to the way one speaks in Irish (níl tart orm or chan fhuil tart orm) – but I believe using the article is much more common and the default in Scotland.
Similarly I am hungry is basically always tha an t-acras orm in Sc. Gaelic while tá ocras orm in Irish, but then some older 19th c. books for Gaelic give examples such as tha acras orm.
Is not the thirst on me is the literal translation. While you will have seen that it is possible to have phrases like 'tha mi àrd' (I am tall) for describing people. When describing transitory sensations such as fear, hunger, thirst, anger, this is the construction they use. There are forms you could use with the other version but they would give the implication that this was the person's natural state and that they were that way all the time, so stick with this unless you have a very good reason to not do so.
It’s oirnn. And yes, it means on us, so it is a plural equivalent of orm on me. The basic preposition is air on and you can find how it’s conjugated by grammatical persons there: https://en.wiktionary.org/wiki/air#Scottish_Gaelic