Why that and not this?
I'd say "This production lasts between three and four months".
I believe there's no particular reason. I would say this as well.
Same issue here
Please explain why dauert is used instead of some for of nehmen?
Because most expressions do not translate literally between any two given languages.
'To take' is not used in all of the same ways as 'nehmen', and vice versa. To express how long something takes or lasts in German, you use 'dauern'.
The speaker's enunciation of 'zwischen' is terrible. She seems to have a very hot potato in her mouth.