So what I would like to do is a screen where users can enter schools with basic information like name, website, and a contact if known.

But I also like to have a section expand if they want to enter more details.
I know you did already some pages that have this little triangle - if you click then the section would expand.

The form - once expanded - should include:
Postal address
Up to 4 contacts (with email address)

Maybe you do a quick sws that inserts the partials: