I wouldn't use soap, just a quick dry burn, rinse, repeat and if your coil is still badly gummed up, build a new one.
I'd worry about soap getting into the nook and crannies of the deck, and unless you take your deck off the mod, the rinsing required to ensure that you've removed all traces of soap could be risky.
Toothbrush can be fine if you're using a gauge that's strong enough, but you may end up having to re-balance your coils after should you move the wraps a bit. I just gently scrape with my ceramic twizzers and that gets the worst bits should they be stuck to the coil.
About dry-burn, the goal is heat to the point that some colour starts, so that the gunk comes off it or ensure it's heating correctly, but never glow them hard... This modifies the metal's structure and create oxides, and neither are good.
I wouldn't recommend doing a cleaning via dry-burning more than 3 times max before you switch to a new coil.
The worse thing I see is anyone that glows their coils red (or even white) for the "cool" factor, and not getting that this is a health risk.
I advise that if you're going to glow your coil for any reason, do it in a dark room and only do it to when you see the color starting.