Finally I figured out powering scheme for segment and can start assembling the thing. For logic I will still use 5 volts, but for segments it will be 9 volts. Here I found some advice for dealing with powering segments: https://electronics.stackexchange.com/questions/233157/driving-large-common-anode-7-seg-led-with-npn-transistor-and-arduino-worked-but. As of right now, the display area (digits) and its powering circuit is complete. Here is how it looks like: