NEMA17 should be OK. There is a big variety of motors that fit the NEMA17 frame size, so I cannot be definitive. If you can, plan on a toothed belt drive with a 2:1 ratio or higher. You can change the ratio later with different pulleys.
Calculate the maximum unbalance torque from your camera and get a motor with twice this holding torque. That is after the gear ratio multiplication, so if you use 2:1 then your holding torque could equal the unbalance torque.
Yes, the Uno will be good for two motors and that LCD. Sound should be a piezo buzzer that just takes an on-off signal from one Arduino pin. Some other LCDs come with 5 buttons or a tiny joystick switch. That is useful to make a user interface. The LCD and interface will be the most time-consuming part of the project.