As said, LEDs are diodes, otherwise called rectifiers. They are DC semiconductor devices, but don't have to run on already rectified DC. As diodes, they do not conduct the negative portion of an AC signal when forward polarized. They drop a specific voltage (2.5 -3 volts typically) so a bunch of them strung together as in Christmas lights would add up to the total line voltage.
I've never had one in my hands to evaluate, but also as said, the bulge in the cable could be a fuse, or a current limiting device such as a resistor, since LEDs are also spec'd with a maximum current parameters.
Because of their nature as DC devices dropping a forward voltage, conventional incandescent dimmers cannot be used unless some sort of intelligent interface is used to track the chopped AC signal from the dimmer and cleanly produce a dimming effect in the LEDs with, probably, a pulse width modulated output.
About the Triac leakage though, it was my understanding that triacs naturally have a leakage current through them that will not let them shut completely off unless there is enough of a load (low impedance). A string of LEDs is a pretty high impedance, as are two-wire electronic fluorescent dimming ballasts. Those loads are tough to shut off, and the best solution is an air-gap relay in series between the conventional dimmer output and the load.
Crestron CLX modules will shut off the LEDS and fluorescents because each dimmer output has a relay at the ass end that opens at 0% dimming. Lutron modules will do it as well, as long as only one of the four outputs is used. However, they are still all incandescent dimmers, and still will not dim LEDs [non-interfaced] correctly.
No doubt that lighting control companies can develop LED dimmers, but Luminary companies try to develop LED lamps that can be used with all that's out there now. I suppose eventually there will be convention, and incandescent lamps will go the wayside of vacuum tubes, as will their controls.
For your shut off problem, you could try wiring a small incandescent lamp in parallel with your LEDs. Maybe even a 7 Watt nightlight hidden in a closet could do the trick. Best of luck.