How to solder or mount this potentiometer on a PCB

You've under-etched the board, or the photo resist was not properly exposed or developed to reveal the copper for etching (so it would definitely be under-etched in that case).

The way it works is that the etch resist prevents the copper from being etched away. When you are done you should be able to see light through the etched areas (and there still should be resist protecting the traces).

A photo would help, but it may be either that you did not recognize the appearance of an etched board or that something is funky with your etchant that is causing the areas being etched to not have a bright matte copper appearance. Here's a partially etched board where you can see the laminate partly exposed and the copper remaining.

With limited information regarding the board and it's mounting hole pattern, it's hard to give specific information.

What I would recommend for enclosing general purpose large boards would be a 1U or 2U rackmount enclosure. They are fairly common for electronic equipment. If you lay your board inside it, you can mark the mounting holes with a sharpee and then drill the holes out. Once you have your custom hole pattern you can use some stand offs to secure it in place. You'll also need a few screws. You might consider using nylon standoffs if you want electrical isolation or the PCB mounting holes are connected to the ground plane. Below are a couple of links to get you started:

2U rackmount chasis (435mm x 295mm x 80mm): $39.95 http://www.circuitspecialists.com/rackmount-enclosure-37-2u.html

PCB heck standoff 1/2": http://www.digikey.com/product-detail/en/2203/36-2203-ND/82512