History is full of instances where superior numbers are defeated through strategy. To give a few examples, the forces of Hannibal were outnumbered by more than 50% at the Battle of Cannae. Yet due to tactical positioning, false withdrawals, and awareness of the tactics of his opponents he managed to destroy the majority of Rome's army while taking only minimal casualties on his own side.

In contrast, at Zama, the final battle of the Second Punic War, the Carthaginians had a slight numerical advantage, as well as several elephant troops. In a straight on battle, the Romans would have been crushed underfoot. Publius Cornelius Scipio, the Roman general at Zama, was well aware of how Hannibal liked to use his elephants, and devised a plan to split his regimens into lanes at the last moment, not only avoiding unnecessary losses against the elephants but allowing them to be dispatched and unable to effect the battle. While the timely arrival of Massinissa's cavalry reinforcements helped, it was tactical awareness that won the day for the Romans.

These are just two examples, and very brief ones at that. Harold Lamb goes into the tactics in greater detail in his book Hannibal: One Man Against Rome, though it is worth acknowledging that his histories tend to have a certain narrative property to them. A more militarily focused historian might give you better insight, but I do not have the experience to recommend one. Still, for basic questions of how strategy can shape a battle, Harold Lamb will demonstrate quite nicely.

To add to what /u/PrincessAnika and /u/NinjitsuSauce have said, "big armies" can sometimes be a handicap rather than a help. Imagine a situation in the 18th or 19th centuries where you have lots of small roads leading to a particular location. Now, if you have a large army, you're going to have to break your forces into multiple packets to moved down those separate roads. If you don't, your army is going to be strung out along one narrow route. It's going to move slowly, because any roadblock ─ say a cannon throwing a wheel or simply the deep mud churned up by thousands of feet ─ is going to slow everyone behind it.

If you break up your force into smaller contingents, you're going to have other problems. If you don't have a telegraph, you're going to rely on runners, and that means you won't be able to coordinate the disparate parts of your army. You can say, "Let's meet at 4 o'clock," but what happens if one of your sub-generals is slowed by that broken cannon or an ambush? Do you go ahead with a smaller force, or do you wait for them? If you wait, what is your enemy doing in the meantime?

In addition, a larger force is going to need more in the way of supplies to stay larger. You'll need more food, ammunition and simple subsistence. That's also going to strain the routes leading to a particular location.

A smaller army, moving along those same roads, won't have the same problems. It'll be able to move more quickly, and it won't need as much in the way of supplies. Led competently, it might be able to attack and defeat a portion of that larger army, then fall back and repeat the process again and again.

The classic example of this problem is the Seven Days' Campaign in 1862. There, the U.S. federal forces vastly, vastly outnumbered the Confederates who were defending practically at the gates of Richmond, their capital. Yet the Confederates won because they were able to strike repeatedly at a federal force that was divided by geography and command.

I am not adequately educated to answer your first question, but I will try to answer the second and third one by examples from late 9th to early 10th century Europe, the invasion of the Hungarians in particular. Background: until around 896 A.D. Hungarian (/Magyar) troops were in frequent employment of European rulers, particularly East Francia, Byzantine and Bulgaria to some degree. After the Magyar invasion of Pannonia, the past alliance between the Magyars and East Francia was not renewed. As consequence the Magyar found themselves at war with the greatest kingdom of Europe at the time, the scales tippes greatly against them.

Yet there are three battles where the Magyars were able to snatch victory from the enemy against overwhelming odds:

-the battle of Brenta in 899 A.D. where 5000 Magyars battled 15000 soldiers of the Italian king Berengar and managed to win, consequently sacking Feltre, Vercelly, Modena and several monasteries along the way

-the battle of Pressburg in 907 A.D. where 60000 Bavarian-German soldiers lost against 20000 Magyars, leaving the majority of Bavarian nobility dead on the battlefield

-the first battle of Lechfeld in 910 A.D., no estimates exist, but the significance of the battle and the threat of Louis the Child to execute anyone who denied joining his army makes it easy to imagine an army at least the size the Germans represented at Pressburg. Louis still lost and agreed to pay tribute to the Magyars.

There are sources regarding the "Western" leadership of the East Francian/German/Bavarian armies but there is sadly little quotage regarding the Magyar leader. Due to his legendary fame Árpád is usually marked as the leader of Magyar troops, although it wouldn't explain why the Magyar military "luck" changed so drastically in the coming decades (2nd Battle of Augsburg in 955 A.D. for example), hence for the standard of accuracy I'm presuming we're talking about an unnamed but immensely capable general who, thanks to his skills in leadership and warfare, was able to repeatedly defeat significantly bigger armies. What were the factors that determined his victories?

-psychological warfare, for example terrorizing and demoralising the enemy with constant, repeated attacks, inflating the enemy's confidence and, lulling his vigilance, with deceiving manoeuvres or false negotiations, than striking and destroying them by surprise attack (Battle of Brenta, Battle of Augsburg in 910, Battle of Rednitz)

-feigned retreat (Battle of Brenta, Battle of Augsburg in 910), -effective use of military intelligence, thus preventing surprise attacks and attacking before all German forces could combine (Battle of Augsburg in 910)

-rapid deployment and movement of units, surprising enemy troops (Battle of Augsburg in 910)

-covertly crossing geographical obstacles, thought by the enemy to be insurmountable, and then attacking unexpectedly (the Danube in 907, the River Brenta in 899, the Adriatic Sea to reach Venice in 900)

-use of nomadic battlefield tactics (feigned retreat; swarming; hiding troops on the battlefield and ambushing the enemy; emphasis on surprise attacks; dispersing military units; fluid, ever changing battle formations; exploiting of the superior mobility of the light armored cavalry; predominance of the horse archery; etc.), the value of which is shown by their victories in those battles

-extraordinary patience to wait days or even weeks for the right moment to engage the enemy and win the battle (Battle of Brenta, Battle of Augsburg in 910) -maintain superlative discipline among troops in respecting and perfectly executing orders

-killing the enemy commander, (Pressburg, Eisenach, Augsburg, Rednitz.) This was also a tactic the Mongols employed, and it served to weaken the enemy by "cutting his head off", as well being an effective psychological effect that made surviving enemy leaders afraid to fight them again

Sources: Johannes Aventinus: Annalium Boiorum VII Chronicon Hermanni Contracti: Ex Inedito Hucusque Codice Augiensi, Unacum Eius Vita Et Continuatione A Bertholdo eius discipulo scripta. Praemittuntur Varia Anecdota. Subiicitur Chronicon Petershusanum Ineditum

Bowlus, Charles R. (2006). The battle of Lechfeld and its aftermath, August 955: the end of the age of migrations in the Latin West

Gubcsi Lajos: 1000-1100 years ago. Hungary in the Carpathian Basin, Budapest 2011 Bóna, István (2000). A magyarok és Európa a 9-10. században ("The Hungarians and Europe in the 9th-10th centuries") (in Hungarian). Budapest: História - MTA Történettudományi Intézete

Györffy György 2002 p. 214 From Antapodosis of Liutprand of Cremona. Hungarian translation from the original Latin

A battle, in many ways, is a gamble. Commanders would try to stack the odds in their favor in a number of ways to influence the battle. Numbers, as you alluded to, is one of these determining factors but there are many more. Things like troop composition (what style of troop made up the forces i.e. light infantry), the logistical situation (how well equipped, fed, and rested is an army), the discipline/experience of the commanders and the men, morale, terrain (including natural terrain or intentionally fortified positions), and even how the two armies met would determine the battle as well.

If a vanguard made up of French heavy cavalry marching ahead of the main body stumbled on an entrenched English position of longbow men (who were capable in a melee as well) and heavy men-at-arms, then this would spell defeat. A French vanguard catching the same English army unprepared will decisively rout it (see Battle of Patay). Numbers won't amount to much if your army is composed of hastily drafted inexperienced and poorly equipped levies or if the leadership is composed of rivaled aristocrats looking to compete for prestige and honor. Large numbers of disciplined men can be foiled by using bottlenecks (Thermopylae). If your enemy fights in tight formations, then use types of broken terrain to make them lose cohesiveness. Ambushing larger armies while they camp (Battle of Las Navas de Tolosa) was a successful strategy too.

At the Battle of Gaugamela, everyone knows that Alexander led his cavalry on a successful attack into the Persian lines, but what about his generals who successfully held off the main Persian assault that included scythed chariots? They got the men to shoot them down from a distance with missiles and then form channels in their lines to allow the chariots to pass through before killing the drivers. They then fought and held off an overwhelming number of Persian infantry and cavalry allowing Alexander the opportunity to attack. Good commanders are complemented by good officers.

In short smart commanders would take into account several facets before and during a fight to ensure victory. Many of these things have as much to do with winning a battle as "out flanking an enemy" or using "a wedge formation" or "hammer and anvil strikes". Many battles are won and loss by an unexpected turn causing morale for one side to dip and the other sides capitalizing on it.

edit for wording