Here is the item again:
While Dr. Kinser bounced down the stairs
(A) and strode through the
hallway. (B)
A brightly colored tie streamed over his
shoulder, (C) flapping
in his wake.
You correctly realized that the period after hallway created a fragment. While Dr. Kinser bounced down the stairs ... is a subordinate clause, so it cannot stand alone as a complete sentence.
To fix this fragment, you chose to do this:
While Dr. Kinser bounced down the stairs and strode through the
hallway, a brightly colored tie streamed
over his shoulder, flapping in his wake.
You have correctly employed Punctuation Rule 2:
Subordinate clause + , + main clause.
Play that music!