What is the case required after от? I think it's genitive, but I'm not sure...
Every preposition requires a certain case, sometimes two cases, and for the majority of them it has nothing to do with the meaning. You just need to remember that от is used with Genetive
Why is it от школы and not из школы? Because I was thinking if you say I "go to school" as "в школу" and i am "in school" as "в школе", then why doesn't it stay in the same pattern for it's location-preposition and use из for use with "place with boundaries?" I thought, у, к, & от were for use with "a person's place, i.e., their house."
The words далеко and от are permanently paired with each other. Such as "far from".
ИЗ (in what you meant) denotes movement from inside of something. ОТ (in this sentence) denotes being at a distance from something. Although ОТ also can denote movement from smth. Я бегу от собаки.
Though i disagree it is more natural, it is an alternative. I don't understand why someone has downvoted you for this.
However, "a long way from here", for example, is a regional colloquialism and is not a universal formation. So, you are colloquially correct, but it is not grammatically sound English.
(I am also a Britsh English native speaker)
от кого ‧ from whom
от чего ‧ from what
откуда ‧ from where
отсюда ‧ from here
It's not a good idea to split "далеко" and "от". So, on it's own "они далеко живут" is fine, but here it's better to say "Они живут далеко от школы". "Они далеко от школы живут" is possible in speech but in writing it looks awkward.