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, which 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!