Is it possible to position all chess pieces on the board such that all pieces defend at least one piece and attack at least one piece?
Yes, and it's not particularly hard to do. Here's one solution.
I found this by placing the pawns first (because they are much more limited in their attack/defence potential), and then adding the pieces one at a time. Using this simple manual approach, it took about 15 minutes to find a solution. As an added constraint, I avoided placing the kings in check.
There are undoubtedly many alternative arrangements that would also work. It would be quite straightforward to write a computer program to search for valid configurations. A simple brute-force backtracking algorithm would probably suffice.
It cannot be done "naturally" in a normal chess game though.
The pawns on columns A and H only have 1 square of influence for attacking/defending, since they are on the edge of the board. To move them further into the board, they would have to capture another piece, and if you do, you don't have all the pieces on the board anymore.