Block cipher's output in case of uniformly drawn input
The answer is yes.
While I can't provide a rigorous proof (because I don't know for now how to make one for such distributions), I can give an intuitive argument, which you can formalize I hope.

The main reason is simple: $E_k$ is a permutation for all choices of $k$.
This means that essentially all the encryption function does is "re-label" the given input message, there are no collisions (which would change the distribution) and input- and output space are equal in size so, every value in the input space is mapped to precisely one value in the output space and vice-versa.

So if you have probability $1/2^n$ of drawing $m$ you have probability of $1/2^n$ of ending up with $E_k(m)=c$ and so $c$ is distributed uniformly at random.