I honestly don't kow
It is a common noun when you are talking about a schoolhouse in general terms. There are also some hotels and restaurants and such places that are called the Schoolhouse. In those instances they are proper nouns.
Go West, Young Man
Schoolhouse Rock! songs
Schoolhouse, skeletal
Classic looking schoolhouse lighting can be found at many retail stores. Home Depot, Home Hardware, each have a variety of schoolhouse lighting fixtures for sale. Schoolhouse Electric also has some available on their website.
Sonic's Schoolhouse was created in 1996-11.
The Grammar Rock songs from Schoolhouse Rock include Busy Prepositions, Interjections!, Unpack Your Adjectives, Tale of Mr. Morton and their most popular, Conjunction Junction. Some of the other subjects that are covered in the series are Multiplication Rock, Money Rock, Science Rock and Earth Rock.
Marble Schoolhouse was created in 1835.
Morristown Schoolhouse was created in 1824.
Jefferson Schoolhouse was created in 1851.
Ames Schoolhouse was created in 1898.
Montrose Schoolhouse was created in 1909.