Hi Brighttail,
You've got a keen eye and good attention to detail, kudos to you.

You definitely don't want the back of the circuit board to come in contact with anything conductive, even painted metal.
There are many solutions to your situation. A piece of cardboard, rubber or plastic cut to the same size as the circuit board to use as an insulator/spacer instead of washers. Wide foam weatherstripping (hardware store) can be used as well, even velcro tape will work (3M brand is best). If you don't like the color of the insulator/spacer material you can paint it, but I advise against any metallic paints (potentially conductive). Another alternative is to cut short pieces of rubber or vinyl tubing to serve as spacers instead of washers. Have their length long enough to hold the circuit board off of the surface in which it's mounted to, this may/does require longer screws.

A tip for prepping the back of the circuit board if the solder points are sharp and or stand out a good bit. Sand (around 180 grit) the solder points down a little bit with sandpaper on a flat surface such as a table. Most quality component manufacturers will have already done this, but some don't. This helps if you use adhesive backed material to stick to the back of the circuit board if you try that method.

As far as placement? I'm unable to help you definitively as every pc combination generally requires different solutions, the best I could recommend is placing it in a location not to be disturbed by moving/accessing other components but still be accessible should you need to change fan cables.