A) From what I have experienced, Diet Pi is constantly changing the way things work. As a seasoned Linux admin, I don’t like the way Diet Pi mangles things like environment settings, so I don’t use it. The instructions on the Gitlab page worked when they were written. But I cannot make any guarantees on whether they work now. You will likely have better luck posting to the Diet Pi forums for support on the auto-login and auto-execution.
B) At the moment, the library browser is a different web page than the now playing screen - they are both embedded using iFrames. So there is no sharing of information between them. The library page will read the cookie, but it will not set the cookie. For the library browser, make sure that a zone is selected. If you are using multiple browsers to access it, make sure you click the circle reload icon at the top when you switch browsers. For a full explanation why, see this post: Roon Extension: Roon Web Controller v1.1.1
The fact that the library is a separate page than the now playing screen is a huge source of annoyance for me as well. I am prototyping a complete rewrite, but it is no where close to being ready to be released. And since I am currently house hunting and have to move before the end of April, I am not sure when I will have the time to work on the next version.