I don't know for sure why a connection with short length of wire is called a pigtail but it was/is common to twist the wires and sometimes would even be loosely braided. Tail of wires is wide at the plug end and would get smaller at the wire connection ends. So it would be something like the hair pig tail which derives its name originally based on a twisted length of drying tobacco.
Do know pigtail is a common term for a short length of multiple wires with a common connector. Including the short 4 pin hookup you sometimes see hanging at the back of a tow vehicle.
You need a fuse because there is an
electrical wire that will now have power that did not before. That circuit needs to be protected by a fuse or circuit breaker in case there is a short in the wiring.
Oddly enough it is the lack of position sensitivity that is the main reason that people will still install a time delay controller rather than proportional. Since time delay just go through a preset delay and preset ramp up of
electrical power delivered to the
brakes when the brake
lights go on they can be mounted any old way or angle. Orientation does not matter for time delay the way it does for a proportional which needs to detect deceleration and respond proportionally to that deceleration.
What can be funny is without a little bit of modification some vehicles with a time delay brake controller won't know the difference between 4 way flashers and brake
lights. So going up a steep hill with a trailer the driver puts the flashers on because they are moving slow and that in effect taps the trailer
brakes each time the
tail lights flash. Which of course slows them down even more.....