Not always interchangeable, but can be.
The middle is the center, or between (noun). "There is a tree in the middle of the yard." "I found myself in the middle of a fight."
The midst generally means that whatever the object is would be surrounded by something. "There was a tree in the midst of the forest." "I found myself in the midst of a fight."
Amidst (adjective) is pretty much the same as midst, but since it's not a noun don't use an article. "There was a tree admidst the crowd."
This is my best guess, anyway.