I know that App Engine has its own datastore. This is great for most cases and fairly easy to used. However, we have a MySQL database that we use for several applications an
You cannot create a direct network connection to your database. The overview page outlines the major restrictions that would stop you from using Mysql - the major one in this case being "arbitrary network connections". You can only make http(s) calls from within app engine.
The JVM runs in a secured "sandbox" environment to isolate your application for service and security. The sandbox ensures that apps can only perform actions that do not interfere with the performance and scalability of other apps. For instance, an app cannot spawn threads, write data to the local file system or make arbitrary network connections. An app also cannot use JNI or other native code. The JVM can execute any Java bytecode that operates within the sandbox restrictions.