The Dyna-s leaves the coils on pretty much all the time. They heat up in ratio to the coil primary resistance. 3 ohm coils draw more than 5 ohm coils, so they will heat up more.
The only time the Dyna turn the coil off is during firing, which is about 20 -40 degrees of crank rotation, and only when the crank is turning. At all other times the coils draw power.
The plugs should fire in pairs, and it needs the mated pair installed to fire along with the removed one you are examining. It uses the head metal to make an electrical connection with its mate.
The coils do NOT need to be grounded. The coil windings, both primary and secondary, are isolated from case or mounting brackets.
I can't speak toward why your Dyna-s trigger wouldn't function. Certainly a new one should, though there have been many reported failures of ones that have seen service. But, that is usually limited to a single coil output (2 cylinders dead).
Cheers,