Does MySQL foreign_key_checks affect the entire database?

It is session-based, when set the way you did in your question.

https://dev.mysql.com/doc/refman/5.7/en/server-system-variables.html

According to this, FOREIGN_KEY_CHECKS is "Both" for scope. This means it can be set for session:

SET FOREIGN_KEY_CHECKS=0;

or globally:

SET GLOBAL FOREIGN_KEY_CHECKS=0;

As explained by Ron, there are two variables, local and global. The local variable is always used, and is the same as global upon connection.

SET FOREIGN_KEY_CHECKS=0;
SET GLOBAL FOREIGN_KEY_CHECKS=0;

SHOW Variables WHERE Variable_name='foreign_key_checks'; # always shows local variable

When setting the GLOBAL variable, the local one isn't changed for any existing connections. You need to reconnect or set the local variable too.

Perhaps unintuitive, MYSQL does not enforce foreign keys when FOREIGN_KEY_CHECKS are re-enabled. This makes it possible to create an inconsistent database even though foreign keys and checks are on.

If you want your foreign keys to be completely consistent, you need to add the keys while checking is on.

Actually, there are two foreign_key_checks variables: a global variable and a local (per session) variable. Upon connection, the session variable is initialized to the value of the global variable.
The command SET foreign_key_checks modifies the session variable.
To modify the global variable, use SET GLOBAL foreign_key_checks or SET @@global.foreign_key_checks.

Consult the following manual sections:
http://dev.mysql.com/doc/refman/5.7/en/using-system-variables.html
http://dev.mysql.com/doc/refman/5.7/en/server-system-variables.html