Yes, one digital IC output can be connected directly to a few digital IC inputs of the same family. That is essentially what your are doing through the switches.
Not sure what you mean here, a pull-up would not help as the problem with the emitter follower is that it doesn’t pull down.
If you only want to generate 5V output pulses from your 12V supply, you could use the same output circuit as in your 4024 based circuit.
The main problem with that emitter follower based buffer is that it requires relatively small resistors which lead to relatively high power consumption. (On the order of 4mA when the output is high which might not be an issue as the LEDs already consume about 2mA each).
I haven’t done the math, but it might be worth it to use the usual opamp based buffer, if current consumption is a concern.