Who would be allowed to operate with no navigation lights over a populated area? I know little about aircraft, FAA or FARS, but I've tried to research something I saw one evening without much success. I was hoping an expert could chime in.

One evening, while watching for satellites, I saw a black blob cross the sky, and thought that was really strange. A few minutes later, it happened again, in nearly the exact same position, and in the exact same direction, but this time I realized it was accompanied by the sound of a prop plane. This went on a few times, each time the sound of the prop plane and then a black blob going south to north with no lights. Eventually, the 4th or 5th time, an SWA 737 was coming in for a landing at a nearby airport, MAF if it's relevant, perpendicular and in the path of the dark prop aircraft, when suddenly the dark aircraft lit up it's navigation lights. I could only assume it did that because of its vicinity to the commercial flight 

From what I've read, all aircraft abide by FAA regulations, even the military in general. Does anyone have any idea of who would be allowed to operate an aircraft, with no navigation lights, over a populated area? My only guess is that maybe federal law enforcement would be able to do so?
Military aircraft actually aren't bound by FAA rules. They have their own set of rules to follow. Having said that, though, their rules are basically "Follow the FAA rules, with a few exceptions." And, yes, one of those exceptions is for lights. Military aircraft can turn off their lights after dark, but only if they're actually flying a mission in an active war zone, or if they're doing training in a restricted area that's been cleared of other aircraft.

According to FAR §91.209(b), "The anticollision lights need not be lighted when the pilot-in-command determines that, because of operating conditions, it would be in the interest of safety to turn the lights off." The "operating conditions" mentioned generally refer to a situation where the glare from the bright, flashing anticollison light reflecting off a cloud would blind the pilot. Note also that this only applies to the anticollision light, the position lights (the steady red, green, and white lights on the wings and tail) must be turned on from sundown to sunrise.

Since you didn't mention the sound of gunfire or bombs dropping, I think it's safe to rule out the "military plane in active war zone" explanation. And since there was a civilian 737 in the area, we can also rule out the "military plane on training exercise" explanation. Thus, I think its safe to assume that this was just some guy (probably practicing his night takeoffs and landings) who forgot to turn his lights on.