Is it possible to reinstall MyBB, but keep the forums, users, and posts? Or no?
Yes it is. Is this because of the problem with the theme selector ? have you tried creating a new default theme and template set
Yes to all those.
And how would I go about keeping the stuff?
I've tried, everything in the link you sent, but not even the default theme shows the theme select option.
What kind of hosting setup do you have ? Are you able to create another database and do a test install of a new MyBB ?
I have godaddy with only 1 url.
Yes, you just have to merge the database with your fresh install.
100% don't reinstall. If you keep all the data you're not reinstalling anything. There are exactly 0 things a reinstall will fix that can't be fixed a proper way.
If it's about the theme select issue, let's keep the discussion there:
https://community.mybb.com/thread-196687.html
However I stress again, there is never a need to "reinstall" a forum.