YOU become the path for the electric current to follow if you're holding onto a damaged cable!

I need 220 volts to run a power tool. Can I use one of the small foreign voltage converters sold online? They are advertised as step up transformers from 110v to 220v, but they seem to be intended for people traveling to different countries.

Well, the first requirement is that the converter should be approved for use in the USA. Next, you would want to make sure the maximum power capacity of the converter is large enough to handle the maximum power draw of the tool you intend to use with it.
Chances are that if your converter is one commonly used for traveling to a foreign country, then it would not work for this type of application.

I have a multi-tap 480 volt transformer with 10 primary leads. After I connect H4 and H7 for 480v and connect line to H1 and H10, what do I do with the remaining primary leads?

Usually unused leads are capped off with a wire nut.

The power cord on my light fixture reads (UL) SJTW 105ºC 300V and the sticker on the lamp socket reads 150W 600V. Can I plug this lamp into a regular power socket? Should I be concerned about the different voltages?

Yes, this fixture can be used on your home power of 120 volts.
What it means is that the cord is listed for use on up to 300 volts and the socket up to 600 volts.

While changing the plug in on my treadmill I noticed a white powdery substance inside the cord. It made my throat scratchy and irritated. What substance is this? Is it dangerous? Why is it on the inside of the cord?

It’s just talcum powder. It’s used to keep the inside wires from rubbing against the outer sheath. With some cords, the outer covering can break down and become sticky. This helps to prevent that problem.