Why is it that almost every time I finish killing the Pendletons and meet Emily at the door, she stares at the posters from anywhere between 5-20 seconds?

just bump into her and she will not stare on them

It happens because you take too much time to get to Emily. It will fix itself once you get better and faster
But yeah, as Korvgyn said,you can run into her to unstuck her if it happens

a more in depth explanation if you want it: