It feels really, really bad to read this over and over again in the past weeks. Man did we fuck up our international image. Anyway. I'm really sorry that you're about to move to Germany while we're dealing with an increase in public xenophobia, I really hope it's going to be a good experience for you! Let's see if I can answer some of your questions:
Can you please tell what regions/bundeslanbs/landkreises in Germany are safe/dangerous for non-white immigrants to live and travel?
We have plenty of Landkreise, so there's no point in naming all. Besides some of the lesser populated areas in eastern Germany, all regions, towns and cities are safe for you to life in and travel to. Even those areas that you call 'no-go' aren't a garanteed death trap for all foreigners. Xenophobia is just more common there, but the violent incidents are small in numbers (doesn't change the fact that it disgusts me as hell, just want to give you a better picture about your safety situation). Yet, I wouldn't particularly advice to start a new life in the deep woods of Saxony. A rule of thumb here would be to advice against regions with high unemployment (with the exception of big cities like Berlin or Hamburg).
Violence towards foreigners or Germans with an immigration background is equally happening in all of Germany. Most incidents occur either in hotspots or in densely populated cities (naturally). In 2014, Police reported that 947.000 people in Germany were victims of a crime, 160.000 of those were people with immigration background. Overall, 16.5 million people with immigration background live in Germany. If you like numbers, you can check the Kriminalitätsstatistik 2014 by the BMI or BKA (not sure if there's an English version though). No Asian country amongst the the top 25 nationalities by number of victims.
So, as hard as this is to say, you'll probably experience racism in Germany this way or another. Hopefully it will 'only' be verbally. There is no garantee to be completely safe, unfortunately. But the numbers say that the chances are very low that you'll get in life-threatening danger. I hope I don't seem cold here, just trying to be realistic.
Does it also apply to Berlin?
While there are xenophobic people in Berlin, they are a very small minority. Berlin is very multicultural and you'll find strong support against any racial discrimination. Hamburg, Cologne and other major German cities are also very open-minded, but differ in culture and life-style.
I like walking in city parks on weekends, going to events and venues. It would be terrible if I felt unsafe or harassed in daily life. In the place I live in now I can walk at 23:00 alone and be safe.
Berlin, Hamburg, Cologne or Munich. Those are the cities you're looking for.
So far, Munich is the only option I have because of its safety and tolerance of foreigners.
Ha. Munich is surrounded by Bavaria though.
Another thing is how people would react to me speaking German with mistakes. Will it make things even worse in addition to being non-white?
Don't worry so much. Everyone in larger cities is used to foreign accents and mistakes. People will appreciate it if you're trying to improve though. Hope this helps. I'd also like to add that the German Police is not your enemy. Don't be afraid to ask them for help in case you feel threatend.