Which Arduino Board(s) is/are right for me?

Well a lot of those motors are just switched on and off, so remote relays or MOSFET switches would probably be the simplest way to drive those (keeping all the high current switching away from the Arduino). Signal cable runs should be shielded or twisted pairs or both.

Are the LED strips to be PWM'd or just switched on/off?

You might benefit from extra input/outputs, suggesting a Mega perhaps.

You'll need to be careful to how everything's powered - separate power for Arduino - and how everything is grounded. Are the motors large?