...does rank truly matter in the civilian world?
It most certainly does, but only within the agencies through which the rank is conferred. I'm not sure if PD and FD rank are transparent between agencies.
When I worked vollie for Menifee Co for a short time, everyone, EMS/Fire, and Sheriff always wore some sort of rank insignia, with severa
l on either agency taking the rank literally.
Perhaps it is transparent between agencies.
It was amusing to see the Sheriff wearing five General Stars on his collar.
Perhaps there may have been some showmanship going on, but perhaps not. There has to be someone in charge. I can easily see how a county sheriff might be that person.
Would a person in the military have to salute or show respect to a LEO wearing such high ranking insignia?
Salute? No, as it's a completely different agency. Show respect? Certainly. Authority is an interesting animal, and takes several forms. I might outrank the pilot, but only the pilot is A-coded on the mission orders, even though I as a navigator might be the mission commander, in charge of a flight of ships. In that case, I would retain authority over the mission as a whole while the pilot would retain authority over the aircraft and the crew, including myself as a part of that crew.
It might sound confusing, but in practice, professionals work things out pretty smoothly. On the rare occasion they didn't, if arguing the point at the time wouldn't amount to a hill of beans, I'd wait until mission completion before raising the issue. If crap was hitting the fan, usually the SRO on board would take the offender aside, along with some words like, "Ok, here's how this works. You have the A-code. You're in charge of this aircraft and it's crew, including me. As the mission commander, I'm in charge of you, this ship, as well as the others. Copy?"
Usually, the confusion was caused by confusion, so furthering that a touch planted doubt, which was good, as then you could quickly reinforce their true role: "You take care of the aircraft and her crew, and let me take care of the mission. Ok by you?"
Unless you clearly outrank an individual (two levels or more) butting heads in the middle of a mission is generally not a very good idea. Happens FAR more often in the movies than it happens in the military. As I said, professionals usually work things out rather quickly.