I am developing a java web application and that includes an applet. That applet is dependent on two jar files:
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You can create an ant script or use Eclipse or any other IDE to automatically package your applet. But your way is correct, too
I wouldn't do these things manually. Finding transitive dependencies is very complex. Maybe darioo's answer is a better way to do this.
This is very common indeed. A couple of hints:
You can always re-build those third party libraries without debug information. That should slightly decrease the size of those libraries.
On the other hand, maybe you shouldn't have a direct connection from your applet to a database. You could create an RMI interface (or something similar) to transfer your SQL and result data to an application server, which actually executes your SQL. This is an important security aspect for your applet, if you don't run this in a safe intranet.