If I connect the pins 6,9,8,12 into the ground then with the other pins attached to 5V (through a resistor) I can light each led in each one of the 7-segment led?
If I want to work with a specific 7-segment digit I must connect the specific pin in to the ground and not connect it if I want to keep it off?
What is the proper procedure for handling the ground pins (6,9,8,18) through my mC Arduino?
You would assign 7 pins as LED segment drivers through current limiting resistors and use 4 pins as digit drivers for this particular display.
The Common Cathodes (grounds as you call them) should be driven 1 at a time using small NPN transistors, (since you could otherwise exceed the abilities of the IC pin and there is no sense over-working it).
You multiplex the display by sequencing each digit very fast in a loop and the human eye "persistence of vision" effect removes the "flashing" you imagine would be there since only 1 digit is ever lit a time.
You have actually made it somewhat harder, using the 595... since you need to add the delay time for loading the 595 to the POV effect. But the concept is the same as in the video above.