Thanks for the advice. The problem should be gone :)

"AI for some reason falls off at the second and third corner, because it turns in with too much speed."
I have also struggled with this. You need to slow down the AI when entering a corner. Find the last blue arrow before the corner: Drag the left handel towards the blue arrow so close that the red line just touches the blue arrow.

The way the AI Line works is that the closer the red line is to the yellow line, the slower the AI drives.