Since you're saying "when," not "after," you're saying the two things happened in the same moment of time. Logically, we may surmise that the action of the doorbell ringing likely happens first and the action of walking to the door was her subsequent response, but by saying "when," you're semantically encasing both events in the same timespan or same moment of time, in the same "when" as each other, the two actions happening so hand-in-hand with each other that you have not parsed them into separate moments in the past but together in the same "when," so you would not use the pluperfect "had rung" but instead the present simple "rung," just like you did for "walked."
However, if you were to say "after" instead of "when," then you would not be including both actions in the same moment but conveying them in two separate past times, past times you are conveying anachronistically. Since you would be semantically first conveying the action of walking to the door and second conveying the action of the doorbell ringing as happening not later, how time normally progresses, but instead further in the past, you would use the pluperfect "had rung," the pluperfect being for when you anachronistically cast a second mentioned action further into the past than a prior mentioned action.
By saying "when," though, you don't do that but associate the two events so closely in time that you have her walking to the door happening "when" the doorbell ringing happens, not after. It's not an illogical thing to say, either, because a doorbell takes a couple seconds to ring, meaning she very well could've started walking to the door when the doorbell was still ringing, like anywhere between the start of the "ding" and the finish of the "dong," and that's if it's a standard doorbell chime and not some musical chime that plays 10, 20, or however many seconds of a melody.