"Her apartment is far from her university."
So first, 彼女の大学 means "her university," which is not what we're trying to say here. The possessive particle の tells us that the next word belongs to 彼女 (the she/her pronoun), so we start with 彼女のアパート to say "her apartment." The は particle come next to indicate that "her apartment" is the topic.
The から particle is used for "from" in a lot of cases. So 大学から means "from the university." 遠い or "far" comes after, to be "far from the university." If you were saying "close to the university," you would say 大学に近い, using に instead of から to indicate being "close TO" instead of "far FROM."
More common yes, but quite a few specific words like とおい and おおきい are always spelled with おお when spelled in hiragana (and such words are normally always spelled in Kanji anyway). Never read a good explanation as to why though! At any rate, given the idiosyncrasies of English spelling where there's any guess how a particular vowel sound might be spelled, you can hardly complain (it's the lack of any correspondence between how words sound and what their Kanji look like that's the real challenge)
I don't think the adverbial form is correct here, you would need to use 近い, not 近く, otherwise you're making it a noun and saying "at my house, a vicinity exists".
If we assume you used the adjective form, then the first seems to mean "From my house, it is near".
I don't think the second sentence is correct though, becoming "at my house, near exists". I think it should be 家の近くにあります ("in the vicinity of my house, it exists"). And then the difference would be more subtle in that the first focuses on the distance from the house, and the second on the location of what is talked about.
(Just a fellow learner, so no guarantees)
How come 大学 is the only one of the different school types that doesn't get 校? I've noticed a few times in this course that when talking of elementary school, middle school, and high school, you have to say 小学校, 中学校, and 高校 respectively, But when speaking of college/university, even when speaking of the actual location like in this question, it's always just 大学, and never 大学校. Why is that?
校 meaning "school" is quite a bit like how in English we also say "elementary school", "middle school" and "high school" but just "university/college" never "university school" or "college school"
大学校 does exist but refers to any post-secondary educational institution, not strictly one that offers academic degrees like a college/university would. It's common in older government-run institutional names like 海上保安大学校 Japan Coast Guard Academy, 防衛大学校 National Defense Academy of Japan, 国立看護大学校 National College of Nursing, etc.