I'm putting a few more hours into my "paladin" character, although that is a loosely applied term because she's also a daedric worshiping cannibal. I tend to do things as honorably as possible, but I did all of the daedric quests in the way that pleased the prince the most, such as killing Barbas and so forth. However, I decided not to join the Thieves Guild. Despite becoming a servant of Nocturnal at the end, I just couldn't bring my character to act as a thief when you don't even find out about the Nocturnal connections until midway through the quest line. So, I decided to do what I usually do and wipe them out.

I turned into a werewolf, ran into the sewer pub and killed Dirge - the only guy who can be killed down there - and then lured the other Thieves Guild members to the secret closet door. I knocked them into the door and they opened it up. I've done that before. The weird thing is, after I went into the secret area and killed every member in there except for Mercer, I attacked Vex and knocked her into the treasury door. To my surprise, she opened it up as well, despite the whole storyline with that door requiring two different keys to open it. It all worked out for my roleplay; 70% of the Thieves Guild is dead and the treasury door is wide open, so Mercer wouldn't be able to cling onto power. Obviously, nothing changes in game, but I can go on thinking I wiped them out. But I digress.

So I'm curious. Can NPCs open up any locked door that they're knocked into? Or does this have something to do with faction settings and/or invisible keys?

Those doors are owned by the Thieves Guild faction; NPCs can open doors that they own.