Translation:There is a big library near my house.
"home" should also be recognized.
Also does anyone else feel like the pronunciation is off for some of these sentences? The "toshokan" part in this one for example sounds particularly strange.
Yeah, for whatever reason it often sounds like she's saying "watoshi" instead of "watashi."
Isn't "big library" the subject of the sentence? In that case, shouldn't it be "The big library is near my house"?
Not a native speaker, but I believe the があります is what necessitates the "there is". For your sentence I suspect です would be used in some capacity.
Lucky so-and-so, our library is less than half the size it was when I was a kid ☹️
why is ookii being treated like a 'na' adjective? would ookiitoshokan also be acceptable?
大きい and 小さい are ones of the few adjectives that can be used both as an い-adjective and a な-adjective. The な form is actually very common, it's pretty much everywhere. That variant you suggested would indeed also be acceptable.