> 1) How do I convert between six digit and eight digit codes?
For questions 1 and 3, Benzene's Game Genie document will tell you. 8 digit codes have an extra byte of information for the compare value. Getting rid of the compare value removes the last two code letters (making a 6-digit code), and also some bits in the third and sixth letters get changed. Read up on the code format to see which exactly.
> 2) Could I (or someone that knows ASM) modify a GG to save codes on power off or modify it to enter more codes?
This is a pipe dream. I don't think it's feasible to do this, unless you both reprogram the GG BIOS to save/load codes from somewhere, and implement a battery circuit to somehow apply power to this buffer when the NES is turned off. (NOT worth the time and effort!)
To enter more codes, you'll have to buy a second Game Genie and connect them in sequence.
> 3) Can you tell me why I need to increse the last letter to the next one every eight while incrementing the value by one? (hope you understand)
Again, read the code format in Benzene's doc. It shows the bit positions for all the letters in the code. With a little inspection, you'll get the answer you need.
A lot of this work is already done for you, so don't try to bang your head against the wall in trying to (re)figure out the GG code format.
So, what is this secret project that you'