1) How long after starting a job is it ok to change it? Should I wait
few months more even if I know I will never be happy in this company?
If you are sure about your conclusion that you will be never happy in this company, a few months of hanging around won't make any positive difference for your resume or interviews.
Find your next job, get and accept an offer, then give your notice. Don't just quit without a new job waiting for you.
2) For sure if I get a new interview from a new company the first
question will be why I am quitting a job after such a short time. How
should I answer and justify this (if it helps, I was 6 years in my
Spend some time before interviewing thinking about exactly why you dislike this job so much.
"Management is awful and is not gonna change" is rather vague and not something you would want to say in an interview. Think about what "awful" means in this context, and how you know that they won't change after being there only two months. Be ready to discuss it when asked.
Be as specific as you can in your assessment. That's the best way to avoid getting into the same situation at a new employer.
You don't want to land in a new job with the same attributes as the last one, and a new employer doesn't want to hire someone who will be unhappy and leave quickly. Once you understand what those attributes are, you can help prevent the next job from being short-term as well.
3) How to make sure that the company I apply for will not contact my
current employer before offering me a position? (do they do that?)
You can't prevent a potential employer from contacting anyone they choose.
I'm not sure why you would care much, but you could always indicate that you haven't told anyone at your current employer that you are leaving and ask that they not be contacted. Many potential employers will respect that wish.
Be ready with at least three references that aren't your current employer.