Student or Learner
Are A of B and B's A always interchangeable?
the stands at the school fair
the school fair's stands
Is there any nuance in their time of using?
Does B's A only apply to creatures instead of to objects?
Your own example doesn't follow the construction you're trying to query. To follow your first sentence and title, it would need to be "The stands of of the school fair".
It is probably helpful for you to know that there's not a simple "rule" you can follow.
The school fair stands were run by volunteers.
The stands at the school fair were run by volunteers.
The stands of the school fair were run by volunteers.
Note that we wouldn't say "The school fair's stands".
The car door was badly dented.
The door of the car was badly dented.
The car's door was badly dented.
I should also point out that "car door" causes great confusion for learners!
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.