I doubt she is going to answer since the website has been sold twice since that post was made nine years ago. Anyhow, you can bypass the switches if you are just going to use the remote, but frankly, I'd not do that. You'd like the ability to to hard switch off the fans for maintenance things like replacing the bulbs. Just install the fan to the switched circuit. Work the switches so that the fan is powered and then just use the remotes. You can put a piece of tape over the switch to remind you to use the remote.

My first thought is the switch is on a switch loop which you did not hook up. If the fan is worked from a pull chain and the light from a switch then you have to have two connections to the power line at the fan. One that goes directly to the fan and the second down to the switch and back again.

Bathroom circuits cannot usually be used for other things except under restricted cases. What else is on the circuit with the GFCI bathroom receptacle?

The bulb in this socket draws enough current that the spurious voltages will not occur. I assume that there are two black/white pairs which come from the breaker box. It's easy to see which black is which. First, connect the white on this socket to the twisted whites and touch the black on the test light to each of the blacks and see if any of the blacks are hot. Once you have confirmed that at least one white/black pair is powered, turn off that breaker and make these temporary connections more secure (not just touching live wires). Turn the breaker back on and then turn on and off the likely wall switches. If none of the switches make the test light go off, then you have found a source of power. Repeat the above for all the black wires and see if any of them are switched. Finally, and I'm not sure about the following test, but if one of the black wires never shows power, it must go to another light fixture or receptacle. Connect your test light between the black power source and this unknown black wire. If your test light glows, then that black wire goes to something which draws current (perhaps a ceiling light with an incandescent bulb in it), and you can figure out what that is by observation. If it doesn't glow, then take another test light, put a regular plug on it, and plug it into each of the likely receptacles. If this second test light glows, then the two lights are in series and you know where this unknown wire goes and you can nut it to the power source wire (or to the switched wire, if you want that receptacle switched). One thing I don't know is how to trace out the white wires. If each of them act as a reliable neutral, then maybe it doesn't matter. If it does matter, I don't know how to identify which is which. Sorry for this long response; I hope I'm clear.