That would depend on what the shield is and how it's recognized by the Linux (if at all). Your Java program will be running as a usual system process on the Galileo's Linux and outside of the Arduino environment, so shield interaction will only be possible if Linux knows something about it, through the facilities available from Linux directly, e.g. GPIOs
Yes, JAR files would need to be transferred to the board to run. That would be outside of teh Arduino IDE again, you'd use something like scp for that.