My parents are moving to a new house, far away from the suburbs, *in a whole a'nother region.

To make it less informal, you could say "to a completely different region"

Looking at my old bookshelves, I felt like returning inI had returned to my childhood: *school manuals, sci-fi novels, a ton of Disney comics.

*Sounds incomplete.

You could say: Looking at my old bookshelves, *filled with school manuals, sci-fi novels, and a ton of Disney comic*, I felt like I had returned to my childhood.

Most of them were in perfect statecondition so I sold them after a quick dust and wipedusting them off quickly.

The slightly damaged ones were to be given to the city library's "donation box".

School makes us buy a lot of manuals which getsbecome outdated very quickly.

And who cares about differential equations when they're not working* in the mathematical field?

*What do you mean to say?

Maybe you mean "they're not being used in the mathematical field"? It is unclear.

It made me wonder how do today's students do manage?

My parents are moving to a new house, far away from the suburbs, in a whole another region.

if you intended to say 'a whole nother region' then this will be interesting:

https://english.stackexchange.com/questions/247785/is-a-whole-nother-grammatical

Looking at my old bookshelves, I felt like returning into my childhood: school manuals, sci-fi novels, a ton of Disney comics.

Most of them were in a perfect state so I sold them after a quick dust and wipe.

The slightly damaged ones were to be given to the city library's "donation box".

And I threw out the rest of the old manuals, not without a sigh.

Throw out = to dispose of something