A tournament
yes they did
Yes
Your question is confusing. You ask about 2 things. War games on horse and hand to hand combat. These do not go together. Please rewrite.
Knights of the middle ages often participated in tournaments of archery and lancing. They also often engaged in hunting, falconry, forms of bowling, golf, and billiards, and occasionally tennis. Fishing was also a popular pastime for some.
Jousting is a traditional war game where knights on horseback charge at each other with lances, aiming to knock their opponent off their horse. It was a popular form of medieval entertainment and a way for knights to showcase their skills and bravery in combat. Jousting tournaments often attracted large crowds and were a key part of chivalric culture.
Attended church, hunted, visited other nobles, played chess or checkers, betting games, raced horses, practiced war techniques, tended estates, wrestled.
Chess is the best-known game with knights and bishops. These titles may be used in other games, also, especially many of the fantasy and swords & sorcery role playing games created in the last 30 years.
It means "role play" or "role playing". So if a game said "Knights RP Game", it would mean "Knights Role-Playing Game".
You don't download games legally. You buy them.
Knights engaged in many types of contests using their skills, but also in the popular festival games of the day. Skill competitions included archery, jousting, and various forms of non-lethal combat. Chess and similar table games were also developed during this time.
Wikipedia's list of compatible games says Knights of the Old Republic II will work but has some bugs. Check out http://en.wikipedia.org/wiki/List_of_Xbox_games_compatible_with_Xbox_360 for details on this and other backward compatible games.