That is actually really helpful in the long run. Different pronunciations, talking too fast, slightly unintelligible... you'll get used to it and you'll get better at understanding it, and if you're ever to visit Scotland and have a conversation with someone there in Gaelic, it wouldn't matter which part of Scotland they're from, whether they speak quickly or quietly, etc, you'll have an easier time understanding them. So, it's a good thing it's not too easy!
His house is pretty different from "he has a house". One is saying what something is, the other is stating a fact. For example, I believe "my house" would be "mo thaigh", but "I have a house" would be "tha taigh agam". It's similar, but it's not the same, so "his house" instead of "he has a house" just isn't correct.
I was inclined to think the same, but think of "my husband", for example: "an duine agam". The thing is, I wrote "his house" as an answer on a whim (based on my understanding of the husband example), and it was accepted. It was, however, flagged for a typo - the app wanted me to have said "hes house" instead of "his". So I'm doubly curious! Looking forward to a native speaker's input!
Probably because you think "tha" means "is" or "are". It doesn't. It doesn't really translate to anything in English. It just shows that something is going on at that moment. It always is used in place of is, has, am, or are.
An taigh aige would be "his house". Tha taigh aige would be "he has a house."