After I invite a teacher, will they immediately get an email saying that they need to write the letter or do I need to assign them to a college first? Also, will teachers be able to school what schools I assigned them to?
The teacher will get an invite from the commonapp.org, but before you invite the teacher and put in his/her email address, make sure you have asked them personally to write a letter of recommendation for you. It is impolite to assign them on the common app and without having asked them first.
This is what is says in the common app resource center re: teacher recs.
“Your online letter of recommendation or evaluation goes to each school to which the student submits, so you cannot personalize the letter. If you wish to personalize your letter, you may send it via mail directly to the school. Please let the student know if you will be sending your letter offline.”