Also:
This
is a glaive.
That thing in the Elf's hot little hand? It's a sword. GW is being helpful and, uh, not sure what they are doing. But its a sword. The things wielded by the High Elf spearmen? Glaive. The Numenorean spearmen? Glaive.
One thing that vexes discussions of realism in Lord of the Rings is that the "olden days" were not homogenic. Things changed a lot, back then. People reacted to things, and tactics and strategies adjusted. What was true in 540AD was not as true in 900AD, and certainly not true in 1340. How armies were raised and who raised them changed utterly between the Roman Era to the post-Roman period to the medieval in England alone. Alfred's standing army looked nothing like Henry's archers at Agincourt. Rohan, for example, are not a cavalry force as a medieval general would understand it - they are rather an army of freemen (who just happen to be mounted), a bit like Harold Godwinson's army in 1066. Only on horses. Unlike Harold's army, they have horses, and they don't suck. Important distinction that. :p
Anyway, heavy cavalry has used swords since the middle of the Roman era, at the very least and probably earlier. Those long, sort of chunky swords we see the Riders of Rohan using in the movie? Those were the Germanic adaption of the older Roman cavalry sword. Wield it one handed and smack it down on the poor [word deleted] you are trying to kill. Lances (as in heavy long spears held "couched" under your arm) were a Norman invention - line everyone up with said heavy spear and charge. The thing we tend to view as the "lance" (long tapered thing to smack against shields) was a much later invention still. And you can't really do the whole heavy-cavalry-with-lance attack unless you have stirrups.
Before that, cavalry armies rather sensibly didn't so much frontal charge as swarm, encircle, skewer with arrows, wheel out of range and rinse and repeat until the infantry army was good and dead. Alexander the Great's cavalry carried these big heavy spears, which they would throw into enemy phalanxes from horseback before things like chariot archers and heavy infantry would come into play.
And everybody had some sort of sword like stabby thing to defend oneself if you were surrounded, or had gotten into close combat.