I'd also recommend removing the paneling. Most likely, the paneling was glue to the drywall beneath it, and although repairing this might make more work, you might be surprised. If the drywall is too damaged by removal of the paneling to repair, install another layer of 1/2". If you want to minimize the additional thickness, use 3/8 and glue to in addition to using screws (2" minimum). I only use 1/4" when I need to wet it and bend it to shape an arch.
For shelving, I never use MDF due to sag issues, regardless how short the span. For less than 32", I've used 1/2" plywood with a 3/4-1" 'L' shaped front edge to give it strength. I usually rout a bead on the lower part to dress it up. As long as you're not storing the Encyclopedia Brittanica on it, it should do fine.
Face frames - if painted - are usually made of poplar. You can either pocket screw them together, or go nuts and use mortise and tenon, or even biscuits, if the frames are wide enough (or you have a faceframe cutter.)