I'm not a teacher.
Hi hokian,
I know the following rules:
1. Participle I Indefinite expresses an action simultaneous with the action expressed by the finite verb.
Having closed the drawing-room door on him, Isabel [STRIKE]a[/STRIKE]waited a little, absorbed in her own thoughts. (Collins) not simultaneous actions
Having reached the classroom, she became the object of many questions. not simultaneous actions
See 2. below. (with "having")
2. Participle I Perfect Active and Passive denotes an action prior to the action expressed by the finite verb.
Mr. Bumble, having spread a handkerchief over the knees…began to eat and drink. (Dickens)
Having already been informed that he always slept with a light in the room, I placed one of the two lighten candles on a little table at the head of his bed. (Collins)
3. It should be noted that a prior action is not always expressed by Participle I Perfect: with some verbs of sense, perception and motion, such as to see, to hear, to come, to arrive, to seize, to look, to turn and some others, Participle I Indefinite is used even when priority is meant.
But it also depends on what other words are in the sentence.
Turning down an obscure street and entering an obscure lane, he went up to a smith’s shop. (Hardy)
I would add 'After' to the beginning of the above sentence.
Hearing a footstep below he rose and went to the top of the stairs. (Hardy)
This one could work. "Hearing" is taken to mean 'Having heard'.
In my humble opinion, your sentence “Taking the key out of his pocket, she opened the door.” is like the written above. So it is perfectly acceptable.
I understand what you are saying, but I think the 'key' sentence goes to far with stretching the meaning. "Taking" doesn't work nearly as well as "Hearing". I would draw the line between the two.
That's why the 'key' sentence clearly struck me as being off the right track.
And I think, for students, it is better to start with understanding the strictly correct grammar.
V.