Caveat: I'm just going on college physics, not electrician experience.
From your description you have three wires running out of your switch (the white one dead ending at the lights) and if it's just a simple switch (one lead in, one lead out) there's a short circuit which would mean your lights would never come on. Or, depending upon how you have it wired, your lights come on AND you have a short circuit. Do you mean the black runs from the lights to ground? If the black is running out of the switch then it sounds like you have a fancy switch with multiple prongs for various types of connections and something is mixed up somewhere that is resulting in a partial short circuit when you turn on your lights. I'd also check your wires and make sure they aren't touching where they shouldn't. You can invest in a simple multimeter ($10 and handy for other things) and use the continuity tester feature to see if your switch is doing what it is supposed to do in terms of opening and closing the circuit.
I don't know much about fog lights but you should be able to read what the wattage is and if your wires and switches can take what's being put through them. From your description of the load specifications the switches and fuses would be enough to let you run a toaster (and you may be doing so if your wires are melting). What is the rating of your wire, especially the one that is melting? The other thing is, even if your wires are officially just OK for the load, you might want to use thicker ones. If you ever run a hair dryer on an extension cord you will notice that the cord is getting warm with time.
A relay is a kind of switch. Basically by closing a switch with a small current passing through it, you cause the relay to close (usually close, though it could open instead if that's what you needed) a remote second switch that has a major currrent passing through it.