How to easily power multiple components with differing requirements?
Well the cleanest option is to have it use a single 12VDC coming in from a wall AC/DC converter. Then you would have voltage regulators to get the specific voltages you need within the unit. Easiest way to do that is to have a custom PCB made, enabling the regulators to be done on board. If that isn't an option, you should be able to get pre made voltage regulator modules from amazon/adafruit/sparkfun. 

I agree with @ChrisStratton that for battery usage, a PI isn't really usable.