Shows and movies sometimes will portray two armies standing to the side as their champions battle to the death. Did this happen? If so, how often?
Yes, it did happen, but I wouldn't say that it was especially common.
One example that comes to mind is the confrontation between Byzantine general Bardas Phokas and the rebel Skleros as described in the Chronographia of Michael Psellos, written in the 11th Century.
"Each side was confident in the face of its foes, and the two leaders by common consent decided to engage in single combat. So, riding out to the space divided between the two battle lines, they scouted out one another and without further delay came to close quarters. The rebel, Skleros, unable to curb his natural impetuosity, broke the rules of this kind of fighting and as he charged, Phokas struck him with all his might on the head. The blow gained additional power because it was delivered on the charge. Phokas, dumbfounded by the unexpectedness of the stroke, momentarily lost control of his horse, but collecting his wits about him, he returned the blow on the same part of his opponent's body. The rebel thereupon lost interest in the combat and turned away in flight."