No, this is patent nonsense done for dramatic effect. From c. 500 BCE AT LEAST, the typically western style of infantry warfare has revolved around two methods of fighting: skirmishing, and shock action. They've generally been combined, although some armies tend to lean more heavily toward one or the other (the early Greek hoplite armies seem to have favored shock action).

Basically, your heavy infantry formed up into block-like formations called variously phalanxes, shield walls, boar's heads, et al. The heavy infantry maneuvered against and clashed with each other in a coordinated fashion. The skirmishers formed a loose screen in front of them to harass and disorganize the other force, or stayed behind or on the flanks to provide archery support.

Greek hoplite warfare, as was butchered in 300, involved more discipline than pure valor. It tended to devolve into a pushing match. Certainly you tried to stab the man opposite you, but as he was armored and behind a large shield, this was not easy. You certainly would not break formation to go after individuals. Indeed, the breaking of your formation generally signaled defeat. Once you were dispersed and running, the enemy infantry could kill you much easier, and more importantly, cavalry could be unleashed on you to devastating effect. Even during the so-called Dark Ages, the evidence is that this style of warfare continued, less organized perhaps than in the classical Mediterranean, and it was perhaps the chief feature of medieval warfare. The battle of Agincourt, for instance, is, rudely distilled, two shield walls clashing in a narrow field, and the compacted French shield wall, wearied from a long walk through ankle to knee-deep mud while under heavy arrow fire, failing to break the English and being slaughtered.

Edit: I should mention that neither the English nor French armies at Agincourt actually used shields, as plate armor had made them superfluous.

Generally speaking, no - when one shield wall had broken the battle was all but won and breaking off into single combat was not only an unnecessary risk, but foolish one as well. To leave the safety of the shield wall when your line has held was dangerous for you and those around you. There was also no compunction about having to face an enemy - during a battle the easier you could remove a potential problem, the better. A great many of the ideals of noble, honourable chivalry (which is generally where this comes from) may have had their roots in the medieval knighthood, but the vast majority were embellished, overstated or outright fabricated by Victorian-era revisionists and the Church.

That said, there are recorded cases of duels being fought before battles - particularly in Celtic traditions and some in the Norse Sagas. Often the duels were ritualistic and taken as a portent of the battle's outcome (which, while not always true does prove something of a point) and would greatly boost morale for the side whose champion won. For the most part this type of duel occurred when two opposing armies, marching to war, encountered one another but weren't particularly hasty to attack.

This type of pre-battle event was all but extinct by the medieval era, however. The Norse formalised their duels into two types: Hólmgang and Einvigi. Both of these types of duels had a unique set of rules to govern them and what they could settle (though this was subject to change). Instead of being limited to champions, any free man could challenge another (though without reason it was generally looked down upon and could earn the byname Hólmganger - such was the case with a man named Bersi), and they were used to settled only very serious matters, or challenge for property and ownership (which was a very serious matter in Norse culture).

In Celtic tradition pre-battle duels faded out as well and the skills they showed were incorporated into other aspects of the culture: feats of strength, agility and dexterity. Displays of combat prowess across the lines are documented as late as the 16th and 17th centuries in the Highlands - sword dances, now an amusing pass-time, were once deadly serious business as they showed the warrior had the speed and skill to dance around a blade, kick it up and flourish it in a manner that would kill a target on whichever side they chose. There's also record of a Graham clansman showing off his skills with a bow before a battle by picking an opposing clansman, reciting a verse (or so it's told) and killing a man thought to be outside bowshot.

So while it's not really as Hollywood often shows it, I think this is much more interesting.

A related question, did 1 on 1 combat between armies occur in Japan often before the Meiji Restoration? Something to the effect of each army's champions battling each other. Did this occur in other parts of the world?

This is from a lecture at my university. The professor mentioned that archers would sneak up on soldiers engaged in swordplay with an opponent and slide a knife under their helmets. The soldier would be unable to focus his attention away from the man in front of him so the archer would simply strike the heel of the knife with a mallet driving the knife into the brain. Occasionally the knife would miss and slide to the side of the neck. If you were lucky you might be able to get away but often the archer would wriggle the knife back and try again.

Unfortunately, I can't recall the period that this type of fighting occurred in. I know this is a weak answer source-wise and apologize for it. I remember the professor's description vividly and have related it as he told.

Are there very concise books with illustrations on formations pre-gunpowder era?

I see a lot of 'Western' explanations here but does anyone know anything about the Eastern way of fighting? Probably the Chinese way would be more representative..

Actually, Spartacus did a good job showing Roman army lining up into its battle formations. scene. My professor for Roman history said the movie did a generally good job of portraying things accurately. Of course, it was directed by Kubrick, so attention to detail is not surprising.

It should be noted that the idealized one-on-one fighting was highly romanticized by Romans themselves, especially in the early Republic before the hoplite army was introduced. After that point it was still idealized but not actually used, and I think it was from this idealization that the Europeans picked it up.