Am I using Java PooledConnections correctly?

This is not the right way. The datasource needs to be managed by whatever container you're running the application in. The MysqlConnectionPoolDataSource is not a connection pool. It is just a concrete implementation of the javax.sql.DataSource interface. You normally define it in the JNDI context and obtain it from there. Also MySQL itself states it all explicitly in their documentation.

Now, how to use it depends on the purpose of the application. If it is a web application, then you need to refer the JNDI resources documentation of the servletcontainer/appserver in question. If it is for example Tomcat, then you can find it here. If you're running a client application --for which I would highly question the value of a connection pool--, then you need to look for a connection pooling framework which can make use of the MySQL-provided connection pooled datasource, such as C3P0.

The other problem with the code which you posted is that the PooledConnection#getConnection() will return the underlying connection which is thus not a pooled connection. Calling close on it won't return the connection to the pool, but just really close it. The pool has to create a new connection everytime.

Then the threadsafety story, that depends on the real connection pooling framework in question. C3P0 has proven its robustness in years, you don't worry about it as long as you write JDBC code according the standard idiom, i.e. use only the JDBC interfaces and acquire and close all resources (Connection, Statement and ResultSet) in shortest possible scope.