And I would not expect any other outcome.
I'm confused as to where the time marker goes.
지금 병원에 있다. Why is the 지금 before 병원에?
Because the child who drives is not a "now" event.
I'm confused too, couldn't 지금 be placed after 병원에 and still be correct? Or am I wrong in my limited understanding of korean grammar?
Because you're modifying "hospital" The child is at the hospital now. You're modifying the child being at the hospital, not "있다".
What is the 하는 for?
운전 is a noun. 운전하다 is the verb. Hence the need for 하는.