Yes , This is very good question from you. I am keen interest to know this answer. If you got answer from any other sources , please post your message.
Yes..... Rest of the code in the doStuff() will available to all the threads. This is the good adavantage of declaring syncronized block ,over declaring method as syncronized. Because , It keeps the entire code in the method in lock mode.
Using the connection polling name , you can create the database connection. You can use datasource to get the connection. If you are not clear refer the document "Getting Jdbc connection using DataSource " in Google Search.