Electrical – Incandescent bulb for 220V vs 110V and difference of their resistance

4.2 microns, not 4.2 microns squared. Area is not a unit of thickness. Divide your answer by pi, take the square root, and multiply by 2 to yield diameter

An 82 ohm resistor would dissipate about 175 watts; indeed, a 15 watt bulb may dissipate that much power briefly. Something the size of a 15-watt bulb that dissipates 175 watts is going to get very hot, however, and when tungsten gets hot its resistance increases by an order of magnitude. You may have been expecting your 1K resistor to dissipate about 1.5 watts, but in practice it would have dissipated about 17.5 initially. Most resistors don't nicely increase their resistance with temperature the way a lightbulb filament does, however, so your resistor may have continued to dissipate that much power until it melted, whereupon its resistance might have actually decreased, causing it to try to dissipate even more power, until things went downhill very severely.

I would suggest that you might consider a Christmas decoration that's designed for a C7 bulb; such bulbs draw about 1 Watt. If you wanted to use a resistor that drew less power, two 33K resistors half-watt resistors in series would probably be reasonably safe if they were somewhat ventilated (the resistors should dissipate less than 1/8 watt each, but half-watt resistors should be used so that in case one fails shorted the other would be able to handle the 0.44-watt fault power) but might not let through enough power to operate the touch switch (if it does work, keep adding 33K resistors until it doesn't work reliably, and then remove one). The fact that incandescent bulbs have a much lower resistance when cold than when heated allows a much smaller energy-wasting element to be used for a given application than would otherwise be necessary.