Is my circuit correct (Nixie Tube Clock)?

Some suggestions about this design:

1. You need more decoupling capacitors and a better understanding of the intention behind their use. There is a capacitor C2 at the output of the LM7805, but it must be connected between the output and GND. In your schematic it is in the current path, between the regulator output and the load (the rest of the circuit). A capacitor does not allow a DC current to pass, so after power on there will be a short moment where the capacitor forwards the rising voltage but this voltage will drop to zero while the capacitor charges.
   This can be simulated here:

^{simulate this circuit – Schematic created using CircuitLab}

Decoupling capacitors stabilize the supply voltage at places where fast switching circuits have a high current demand for a short period of time. The influence on other circuits will be reduced.
The time domain simulation shows such a problem here:

^{simulate this circuit}

VCC1 shows a "dirty" supply of an IC without decoupling capacitor, VCC2 shows the effect of a capacitor close to the chip. It is important, that there is not again a long track between the IC and the capacitor.

It is good practice to place a capacitor from 100 nF up to 1 uF close to each chip between VCC and GND. You can try later to remove some of them, if you want to know where stability issues start. One issue may be noise in a radio next to your circuit.

This is also important for the 4 RGB dots.

2. Shift register connections

• You don't need the shift register Master Reset (SRCLR#) inputs. Connect them to VCC.
• Connect Q7S (QH#) of the first register to DS (SER) of the second to transfer the content.
• Both registers need the signals RCLK (D8) and SRCLK (D7)

3. You don't need R9. R8 limits the current through the LEDs. 1 kohm for R8 may be too high for two LEDs, I suggest 220 ohm.