Electrical – The circuit, especially 7 segment parts does not work at all

As usual, you need look no further than the eevblog exploding capacitors video

AND, OF COURSE, AS A WORD OF CAUTION:

-wear safety glasses

-keep flammables away

-test out everything first with a proper DC voltage supply that has effective current limiting and safety measures

-do it in an actual lab that has real safety and hazard precautions

Not following any of the above is probably not smart [but probably how I would do it anyway ;).] With a low voltage, the biggest danger is probably to actual projectiles.

I would just reverse bias a low voltage cap (for the most flare up). You can put a diode somewhere after it that will 'turn off'. I would also put a current limiting circuit at the input to kick in after the cap goes, for safety.

You could probably get some nice smoke from any IC that you overvolt with a decently rated supply current.

You may need more than a single AA or +9V to get this effect.

According to your schematic, you have the outputs of all counters connected in parallel, and to the anode inputs of your display. You have all cathode leads of the display grounded. This will force all digits to display the same number.

The display you have needs a multiplexed input. You will need seven 4:1 multiplexers between the counters and the display, with the segment outputs of counter 1 going to input 1 of the multiplexers, counter 2 to input 2, etc. You also need to arrange to ground one cathode lead of the display at a time, switching the mux and cathode selection in sync, so that display 1 will be enabled when the mux selects counter 1.