I didn't feel like going out but I had to do.

Or "I had to do it" - "do" is transitive here and, if you include it, needs an object. But the sentence works without "do".

In fact, I live ion thea mountain.

"the mountain" is okay, especially if you feel like it's obvious which mountain you mean - if you live in a place with only one mountain, for example. But "a mountain" feels more natural here.

"in the mountain" - you live in a tunnel that's been dug deep inside, underground.

"In fact" sounds a bit strange, there doesn't seem to be a clear reason for saying it.

Every time I leave home and go to somewhere, I need to ride on my scooter because of itsit's such a steep and long way.

"Go to somewhere" is okay but the "to" isn't necessary and it sounds a little more natural without it.

I see why you'd use "because of", and in a different sentence it would work well. Like, "every time I leave home I need to carry an umbrella because of its frequent rainstorms". But here it sounds strange.

I don't like that so much, but today that's different.

Not sure what you're trying to say here. Maybe "I don't *mind* that so much, but today was different"? Contrasting between how usually it isn't so bad, but today it was bad?