How to drive segment display and sens button press at the same time? Understanding this schematic
The p-n-p transistors Q3-Q5 work in the so-called "reverse active mode" where the collector and emitter are swapped. The advantage of this weird connection, as far as I can remember from 80s, is that the collector-emitter voltage of the saturated transistor is lower than in the case of the conventional active mode connection. The cathode segments should be controlled through additional resistors in series.

The buttons S3 and S4 are read when the according SR signal is low - if the button is pressed, the common button point will have low (0.7 V) voltage. Of course, this point should be "pulled up" by another resistor connected to +Vcc. Buttons are separated by the D2 and D3 decoupling diodes.