František hledá svoje auto is right too, isn't it?
yup. Such translation is also accepted
I am lost - why not Jeho auto - Thanks
Jeho is accepted as well. But it is normally much more natural and better style to use the reflexive possesives if the object belongs to the subject of the sentence.
Is it possible to say "František své auto hledá." ?
Yes, it answers what he is doing with his car, not what he is looking for.
Thanks! I wanted to know if it's acceptable. But it obviously changes the meaning somewhat.