If I worked for a school program a non-profit organization, do the new employer allowed to call the school or does it have to be the Refrence provided meaning the non-profit Organization?

no

It depends on the school district policy. Most school districts will not allow this unless the profit goes toward the school. My advice is to check with an administrator and ask.

Gibbs college is a for profit private school. This is as opposed to most state colleges which are not public schools and not working to turn a profit.

yes it is