I don't know Sumobot, but in terms of maneuverability, you can't beat a 2 wheel differential steered platform. If the powered wheels are on the centerline it can turn (spin) in place (zero turning radius). In terms of speed they could be the same. In terms of stability I would have to give that to the 4 wheel bot.
Use the Servo library that comes with the IDE. Connect shutter to a PWM pin, 5V and Arduino ground. Send a Servo.writeMicroseconds(2000); to actuate shutter and Servo.writeMicroseconds(1000) to turn it off with a delay in between to allow the camera time to get the photo.
Powering the servo from the Arduino (USB) can cause problems. Try powering the servo from an external (like 4AA batteries). Servos can draw enough current to cause a dip in the power to the Arduino to reset it (and then it will wait for a new button press).
You can assign any digital pin to any LCD input, but you must make sure that the correct pins are matched to the LCD pins in the constructor. I have done this in the past to make wiring from a standalone 329 to the LCD easier.
http://www.pololu.com/product/1444/ Is this the motor and encoder that you have. The output of the encoder is NOT PWM. It a series of pulses at 64 pulses per revolution of the motor. Depending on motor speed the pulses will be very brief. Usually the pulses are fed to an interrupt pin and counted for a fraction of a second then the count and gate times (fraction of second) are used to determine motor speed.