How would i go about finding out if there is the needed driver on that computer where the digispark is plugged in
I can't think of any way to do that. That's like asking if a keyboard you plug into a USB port can tell if you have Photoshop installed or not.
Is there any way to add my program made in c# to the digispark ...
No. The Digispark is a tiny computer based on the Attiny85 which has 8k of program memory, some of which is dedicated to a bootloader which lets you upload new code.
Your project is far too ambitious for the level of hardware you are proposing to use. In fact this video which I watched the other day comes to mind: