By the way I know that they even invented a gender-neutral pronoun instead of he/she. Could anyone explain me please to what extent and how it is used, if at all?
You can read more about it here.
I do not understand the rule for when to use 'om' .
You're probably thinking about the verb tycker om. Here's a helpful post: https://www.duolingo.com/comment/5892480
Thank you. That's exactly what I was thinking.