Art stores and craft stores (Michaels, JoAnns) sell sheets of very thick mat board in different colors and textures and some are metallic. Try cutting a background for a puzzle leaving a 2"-3" border after the puzzle is glued onto the mat board.
First, take a black marker and "edge" your puzzle picture. This will eliminate that white/gray line on the edge of the completed puzzle. It will also give a small illusion of depth to the picture.
You could center the puzzle on the mat board, glue it down, then hang it.
For hanging, hardware stores sell packages of picture hangers that you can stick on the back of the matted puzzle picture (use two, one on each side of the picture), run a length of picture hanging wire through the two hooks, twist the ends shut, and hang from a hook or nail on the wall. Ask the store person which hook would be best for your project as your puzzle pictures may not all be the same size/weight.
You could hang them from rods using loops of fabric attached with hot glue to the back of the frame.
For more visual interest, instead of centering the puzzle picture on the mat board, cut the mat board about 1/2"-1" longer on the bottom. When you place the puzzle picture on the mat board, raise it 1/2"-1" from the bottom. You should still have the same amount of mat board visible on the top and sides, but a little more on the bottom. Makes the presentation more interesting to the eye.
For even more visual interest, layer mat board in contrasting colors or the same, under the puzzle picture. Example: you've cut the mat board 3" wider on the top and sides and 3 1/2" longer on the bottom. The smaller board would be cut leaving 1/2" wider than the picture all around. The next mat board would be cut 1" wider than the picture all around. Then place the 1/2" wider board on the center of the 1" wider board and glue it down. Place the puzzle picture on the center of that double-layer mat board, and glue it down. Then take the puzzle picture that is now glued onto the 1/2" and 1" wider board and place the top 2" from the top of the single mat board, and 2" on each side. The bottom should be 2 1/2" from the bottom of the single mat board. Now your puzzle picture is raised from the surface of the bottom mat board and it gives you more visual depth and interest.
If you have a very steady hand, you can put a beveled edge on mat board....or the art stores sell a cutting tool that does this.