How to learn about a department's culture? I recently interviewed for a teaching position, but I have concerns about the department's culture. First, I learned that the department adopted a different hiring process for me. They made me jump through some hoops, but then they fell off the face of the earth. I haven't heard anything for 20 days!  Second, I learned that the department is in desperate need of new lecturers since the newly-hired experienced faculty member (established scholar!)  who had moved across the continent to work in this department left. I want to contact the-above mentioned scholar and ask him why he left, but I don't know if that's professional. What do you suggest?
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...?"