I can't find how to change the time zone setting on this forum or main website. How do I do it?

you mean change the hours that shows up with your comments ?
if so i don't think you can change them, both main site & forum are based in Europe and show their time zone

on the site the date/time are hard-coded with the server time
one of the numerous thing that can be improved in the next version (and already partially done)

on the forum there does not seem to be an option for changing that

Won't it be a problem to have answers sometimes given before questions, if everyone writes with a local time-zone?

Entries are sorted by id (which is fixed when creating the new comment), not by date. This is visible in the few cases where I manually moved a comment to the top of the list.

And even if they were sorted by date, when using a local time display, dates & times are stored as UTC in the database: the change is applied only when rendering the page for the user, according its preferences.