It appears I solved all the problems... hurray!

4: Why the firmware installation is freezing and how to properly install my "P802NV20.hex" one?
-> The problem is a compatibility with my computer. So I used another one and everything went fine... As I won't update the printer every day, I considered the problem solved.FIXED

6.4: Why have I the "avrdude stk500_getsync() not in sync" error and how do I reach the board?
-> see point 4 FIXED

7: Why is it possible to reach the printer using repetier host/server and impossible with cmd/arduino IDE?
-> see point 4 FIXED

But other ones appeared... the lack of rigidity of the overall assembly gives a big load to the motors of the Z-axis. So it was tweaks and tweaks... I am not fully satisfied but it is working so far. And also the metal bed is a pain in the a**! Not sticky without blue tape, too much with blue tape...