Configuring error logging in MySQL

Is there a way to change what errors are logged by mysqld?

There is an authorization problem with one user on a server with several
heavily used databases. According to dev.mysql.com, the "error log" only
logs critical errors, while the "general query log" logs *every* user query.

If I activate the query log, it will be filled incredibly fast - is
there a way to choose what errors will be logged? I only need to record
connections and login attempts.

Also, can these things be changed without restarting mysqld?