Hi Carol,
Thank you for highlighting an issue which we should all take seriously for ourselves, but particularly so for anyone we are teaching - especially young people.
Today's synthetic cords pose a risk that has never before been part of knotting. The compounds released when nylon (or pp or any of the synthetic fibres) are heated excessively should all be considered to range from harmful to outright toxic and even carcinogenic. While fusing the occasional end of a synthetic cord will pose little more danger than a lung full of diesel engine fumes, we should be mindful
not to do this in any quantity in a closed space such as a room, and we should be mindful of the advice we give to others, particularly as they could well have breathing problems as you describe.
The solution to the problem lays in Lindseys quote
When exposed to excessive heat, nylon can decompose and may release small amounts of caprolactam, volatile oligomers, carbon monoxide, nitrogen oxides, nitriles, ketones, and a wide variety of carbon, hydrogen and oxygen compounds of varying chemicals.
and the key is in the word
excessive.
Simply melting nylon or PP does not create any of the hazardous compounds associated with burning i.e. 'excessive heat'. A flame, even the relatively cool flame of an alcohol burner, is way too hot and easily decomposes these polymers into their dangerous by-products. Often the electric cutters used 'in stores' are also too hot, they are rarely thermostatically controlled and when there is no rope present to cool the element, they rapidly got 'too hot' and burn the polymer left sticking to the element.
PatDucey's method of using a soldering iron is in many ways ideal - these are thermostatically controlled and can be set to melt the end without any fumes. Unfortunately they can be expensive and not everyone is going to be able to run to that expense.
However, there is a solution which is easy, inexpensive and SAFE. It relies on the fact that you do not need to put the cord right into the flame in order to melt it - remember, the flame is always much too hot and will always burn and decompose some of the polymer.
I prefer to use a gas cigarette lighter, it gives a small stable flame.
Light the flame and bring the end of the cord close to the side of the flame. When the cord is still about half to one mm away from the flame, the fibres will start to melt.
Gently heat the end of the cord this way until all of the end has melted over.
Dowse the flame and dab the melted end onto the side of the lighter - it will stick.
Quickly drag the cord away from the lighter, this will drag off any blob from the end of the sealed cord and neck the fused polymer into a taper. It should look something like this:-

If you have melted the fibres carefully, there should not be any discolouration and if you pulled out the blob quickly, there should not be any thickened rim around the end of the cord.
Trim off the whisker and the job is finished.
It takes a little practice, but after a few trials you should have the hang of not getting the cord too hot.