Supply topology for microcontroller and ADC/sensor with noisy input/outputs
Only answering part of your question:

Since the worst noisemaker here is the multiplexed LED display, you have several options:

1- Use a LCD instead, with a constant-current LED backlight.

2- Keep the high LED currents inside a tight loop close to the 7-seg modules. To do this you would put the 7seg drivers close to the display, route the return currents in traces (don't let them into your ground plane), and use local decoupling to close the loop, like several hundreds µF with lowish ESR like 0.1R (Panasonic FC for example). Then, power the whole display from the switching regulator, not from your clean 5V. You could also add a cheap slow LDO or a RC filter to smooth the current drawn from the supply.

Also you won't get 1LSB accuracy from the micro's ADC, the TL431 is only accurate to 1% too, so this feels like you need to clearly specify what kind of accuracy you require...