Unlocking the pro

I started a few free courses and then completed them. Now if I unlock the pro mode would I be allowed to do the projects and take the quizzes for the already completed courses.

yep, its just additional content. So the quizzes and freeform projects come available, even for courses you already completed the free part of

it could be a good strategy, first do all free content and then update to pro to do the rest.