If you are just trying to get a firm sole for a powerful pedal stroke, check out Five-Ten shoes Mountain Biking Footwear | Five Ten. To go less-expensive, use trail running shoes. A very firm one is the Saucony Guide Trail 3. (Old model, but at Amazon). With Saucony, you want an outsole of XT900, not the softer XT600.
As far as going with Frogs, they are great pedals. I have been using the Frog Ti on my road bike for years, and they are perfect, but you should have a better reason than bad shoes to switch to clipless. They will almost definitely lighten up your bike. Are you trying to upgrade the components, to lighten it? The frogs will also give you almost unlimited ground clearance on rough surfaces and in corners.
If you're trying to turn the bike into the lightest it can be, they are an excellent choice. They are more of a performance pedal, though, so unless you are competing or seriously training on your single-speed, there are probably better choices.