The RX part will never work, there's a missing Pull-up in there.

Easier to use a ULN chip and skip the Shift register for a single display. For more than 1, still possible to use I2C I/O extender, or even some dedicated led drivers from Maxims.

I doubt that everything would fit in a 16C54 or 16F627. Maybe possible with a 16F628.

How many different 7 segments display you plan yo use?