This has been established in the new Engaland colonies

No, the southern colonies did not have the first public schools. The first public schools in the American colonies were established in the New England colonies, such as Massachusetts and Connecticut, in the 17th century. These schools were primarily established to educate children in reading and writing.

both cases established limits on public schools' actions based on the first amendment

The first public schools in Alaska were established in 1905 while the area was still a United States territory. The first school in the area was opened in Ketchikan.

In Mexico City. The very first university was founded in 1551, known as the Royal and Pontifical University of Mexico.