Well I don't have the answer to all your questions, but I can explain the seemingly odd behavior of your fan relay. The way a relay works is that the internal switch is activated when there is a difference in potential between the two pins. If either side is disconnected, you get nothing. If both sides are grounded, you get nothing. If both sides are fed 12V+, you get nothing.
When your ignition is on, 12V+ is supplied to one side of the relay by the ST-RUN circuit. When you ground the other side, the relay activates and turns the fan on.
When your ignition is off, one side of the relay is presumably being grounded by the unenergized ST-RUN circuit. When you apply 12V+ directly from the battery to the other side, the relay activates and turns the fan on.
The fan doesn't care whether the ignition is on or off as it is connected directly to the battery. Only the fan relay cares.
As for your switch, you're going to have to mess around with it a bit. You can't wire it as suggested by the directions if you want it to supply a ground. If it is a three pin indicating switch being used in a typical configuration... pin 1 is hooked up to the 12V+ source, pin 2 is the output, and pin 3 is hooked up to ground to make the light bulb turn on when activated. You should simply be able to hook up pin 1 to ground, pin 2 to the output, and pin 3 to a 12V+ source to make the light bulb come on. This won't work with a LED switch, but it sounds like you've already figured that out. I'm guessing based upon your voltmeter test results, you have pin 1 and 2 (input/output) reversed.