You've got a couple of issues here.
First, who ever installed that neutral on the switch twisted the wire the wrong way. The wire should always be looped clockwise, or in the same direction you'd turn to tighten the screw. That way, when you go to tighten the screw, the wire is pulled closer, rather than pushed farther away. That is rookie electrical knowledge. That tells me someone who didn't know what they were doing wired this up.
I'm not sure those white neutrals should even be on the outlet but I'm not an electrician so don't take my word for it. However, you also mentioned "continuous outlets in the next room over lose power as well" which you did not describe in your original question but I can only assume is an additional outlet branched off from the outlet in question that loses power whenever the outlet in question does. I'm not sure why that is an issue or what you want to change about that, if anything.
Sounds like you want to wire up the outlet in a split configuration.
If this were my house, I'd call an electrician, hear me out.
I know everyone online tells you to hire an electrician to change a light bulb or you'll burn your house down. That is not what I am saying at all. Normally this type of job is something home-gamers like us can tackle and save a ton of money.
The reason I'd call an electrician is back to my first point. You had a home-gamer that didn't do his homework make changes to your electrical system. Just because everything is fine now doesn't mean there aren't all kinds of abominations hidden in the wall that you cannot see. I would hire the electrician to not only sort out the switch and fan issue, but to inspect the rest of my wiring and make sure who ever worked on that switch before you only made the simply fixable error we can see. Otherwise you may be in for adventure time every time you open up a junction box. Not to mention if you ever want to add home automation to your house a working neutral wire is pretty essential.
Good luck to you regardless. Stay safe and use a non-contact voltage tester.