oh it does sound normal to me - stressful for you, but well within what's normal for a feisty 3 yr old!
I think it could well be that as the third child she has to be a little louder and a little more 'out there' to get attention, and perhaps that's coupled with you not being able to be as 'on top' of her as you would be if she was your first?
I hope you don't mind me saying so (offended someone on a thread earlier without meaning to so am being paranoid today ) that maybe you are setting her and yourselves up to fail by trying TOO hard to regulate her behaviour when she's having time out in her room.
You almost 'move the goalposts' for her - if she carries on strangling chickens (bless her!!!) then she is told she will go to her room etc; THEN is added on another proviso - that she has to be calm for 3 minutes. I think that's a big, big ask and may just be making an unecessary battle for you all.
My own view was that the time out is the consequence; once the time is done, that's it. You can't MAKE her calm. She will calm when she is able. I didn't expect a calm ds to come out of his room if I did give him time out, I just expected him to have experienced that consequence.....I tihnk even if they come out still ranting, that is them just saving face and giving vent to the frustrations you feel when you're a powerless being!
I also think it must be very hard for her to be told to be quieter....she may literally not have that capacity yet; some children just are loud. Again, a consequence can be applied if it's being done at end of tether level just to annoy you, as in time out in her room.
I think my approach would be clear consequences to actions such as breaking someone else's toy coupled with really, really picking your battles, and using lots of distractions/games/challenges to get everyday things done
HTH