In terms of Coop runs, does that mean that if I were to run the category with my PSN friends who aren't registered on speesrun.com, I can submit a run under only my name?

You can associate the run with your friend's PSN as well (which you should). They won't need an account for you to input their name. If you look through the leaderboards (mostly Trials), the names that aren't colored are non-existing users that were in the run.

Give credit where credit is due.

You can also include them in the description of the video too.