I know there was a similar post like this made about 5 years ago, but that post was regarding 14th century and onwards. Because apparently the practise of knights fighting afoot started in 14th century, but I want to know if 13th century English knights fought on foot too. Like in Battle of Lewes, Battle of Evisham, Battle of Falkirk etc. Because in the illustrations/manuscripts of all knight battles (including 14th century and onwards) I always see the knights on horseback. Obviously if they got knocked-off their horse or if their horse was slain they would have to fight afoot, but did 13th century English knights fight afoot commonly/occasionally? Or only when they had to? And if so, how often did they get knocked off their horses?
This answer focuses on the Anglo-Norman period, so earlier than the 13th century. But since OP was interested in English knights fighting on foot before the 14th century, I guess this will still be of interest.
In "Warfare under the Anglo-Norman Kings 1066-1135", Stephen Morrillo argues that combined arms tactics was central to many battles in this period, and also points out that sometimes dismounted knights played the part of the heavy infantry. In particular, dismounted knights appear in the Battle of Tinchebrai (1106), the Battle of Brémule (1119) and the Battle of Bourgthérolde (1124), although often along mounted knights and other infantry. Although the example of three battles seems to be quite small, considering the fact that battles were rare in this period (Morillo counts 7 battles, C. W. Hollister counts 5 battles of significance), this actually accounts for a significant number of Anglo-Norman battles.
So why did this dismounting occur? A knight on horseback can easily retreat, while a knight on foot has less chance of escaping a rout unharmed. Forced into this situation, a dismounted knight must either conquer or die. Indeed, this is often the reason given in the sources for the dismounting of knights. Dismounting has a psychological effect, helping to counteract the fear of harm and death that pervaded every battlefield. In particular, having the knights, the military elite, join the battle lines on foot seems to have been massively encouraging to the common infantry as well. Leaders would often be part of the dismounted to further boost morale. For example, King Henry I seemed to have been on foot in the Battle of Tinchebrai and Brémule, and at Bourgthérolde, Odo Borleng was dismounted with the knights under his command. In this case, dismounting can be done to reinforce and stiffen an existing infantry line, instead of making up the entire infantry line themselves.
As for how the Anglo-Norman army was able to put this into practice, Morillo argues that it stems from the relatively more centralized nature of Anglo-Norman government. He argues that the superiority of medieval cavalry actually stems from the inferior quality of medieval infantry, which requires cohesion brought about by a strong military tradition and/or professional training, both of which were hard to provide in medieval Western Europe. However, due to the more centralized nature of the Anglo-Norman realm and the professionalism of the Famillia Regis (the king's household), there was a strong core force of knights who could function as effective heavy infantry when dismounted.
On the other hand, in the article "Battles in England and Normandy, 1066-1154", Jim Bradbury argues that the dismounting of knights wasn't unique to the Anglo-Normans, but was in fact common in Western Europe. He cites several earlier examples of dismounted knights (such as the Battle of the Dyle in 891) and argues that it grew out of ordinary Frankish tactics, as part of the combined arms response (along with archers and mounted knights) to the threat of a cavalry charge.
This suggests that the practice of dismounting was not uncommon amongst English knights before the 14th century, an might not even be unique to the English. Dismounting was done as part of a wider combined arms battle plan, and deliberately organised. In these cases, the knights would've fought on foot even when not knocked off their horse, since their horse might not even be on the battlefield to begin with.
Sources:
As /u/darthturtle3 has written, knights fighting on foot was very nearly the rule in English warfare from the mid 11th to the mid 12th centuries and was probably part of a larger trend that dates back several centuries. The Anglo-Norman knight, as were their contemporary counterparts and ancestors, was a highly versatile warrior who chose to dismount or charge according to the circumstances he faced.
However, by the middle of the 12th century, this had changed substantially. The Battle of Lincoln, in 1141, is probably the last battle of any significance where substantial elements of the army dismounted to fight, though there were obviously some exceptions to this rule as circumstance dictated. However, on the whole, the trend was overwhelming towards knights fighting battles exclusively on horseback.
Why this occurred is much harder to answer. The tournament began some time in the very late 11th or early 12th century, and reached the height of its popularity in northern France in the 1160s to 1170s. The increased focus on horsemanship and the change in tactics from multiple styles of lance use (over head thrust, uncouched underarm thrust, throwing and couching) a single form (couching) that required considerable training to use properly, in addition to the increasing chivalric tradition, may have reduced the prestige and focus on fighting on foot in general warfare.
At the same time, there was increasing use of town militias for infantry, and some of these forces became quite skilled at warfare during the period. The arms requirements for service in a town militia, combined with the increased number of men who could afford the equipment due to the increased money economy and overall economic boom may have resulted in sufficiently large bodies of well equipped infantry that dismounting knights to supplement their numbers was no longer considered necessary. Civic pride, the close relationships between the townsmen and the use of town banners could also have increased their willingness and ability to stand up and fight that the morale boost of having knights in the line with them was considered unnecessary.
However, these are just theories. We have no clear indication of why fighting on foot was abandoned by knights, just that it occurred. In all the battles fought by the English in the 13th century, knights did not dismount and fight on foot, except during sieges. Even when faced by Welsh and Scottish pike formations, English knights preferred to stay mounted and try to either charge in the hope that the enemy would dissolve in panic or wait until their archers or crossbowmen had thinned the enemy ranks out enough that a charge became viable. It wasn't until after the Battle of Bannockburn in 1314 that the English knights began to dismount with any frequency (such as Boroughbridge in 1322).
This was not yet universal, though, and the initial intention of the English during the Weardale Campaign was to charge the Scots on horseback. Only the treacherous terrain and lack of room to make a charge cause the English to dismount and prepare to fight. Though no battle was fought, this suggests that fighting on foot was not yet the primary mode of fighting for English knights. It wasn't until the Battle of Dupplin Moor in 1332 and Halidon Hill in 1333 that we see dismounted men-at-arms as the primary mode of warfare for the English.
As for your other questions, I'm afraid that I'm also not able to provide an adequate answer.
References
The Great Warbow, by Matthew Strickland and Robert Hardy
Tournaments, by Richard Barker and Juliet Barber
The Art of Warfare in Western Europe, by J.F. Verbruggen
The True Chronicles of Jean le Bel, tr. Nigel Bryant
Infantry Warfare in the Fourteenth Century, by Kelly DeVries