How to learn about a department's culture?
I don't see a problem with contacting him, though his reasons for leaving may not be relevant to your situation. There seem to be a lot of red flags here, but it is possible that they are false alarms. 

I would send him a note saying that you are under consideration for a position at the given department and asking if he has any advice he is willing to share with you. Give a sentence about your own experience or situation (recent graduate, associate professor, or whatever). Any response will probably be useful to you in figuring out your dilemma. 

However, 20 days doesn't seem terribly long and might indicate some turmoil that is being resolved. Don't jump to conclusions, but stay aware. 

If you have any good feelings about any of the people you interviewed with, you might send a similar note to them. "Any advice...?"