The answer to the first question comes from www.tetrabrazil.com:
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”With the advent of a national association in England, any soccer played under its jurisdiction was called association football. As time passed the word association was abbreviated to assoc., which eventually gave way to the word soccer, the game's common name in North America.”
The reason football dominates American college games is found at
http://www.soccer-for-parents.com/us-soccer-history.html:
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"The College Era, and Rules Consolidation, 1862-1875
Collegiate play resumed on a regular basis after the Civil War. Rules varied widely among different schools and communities; Princeton played with 25 players, some people even played a game with innings, with a victory going to the first team to score a fixed number of goals (a la volleyball). In 1866, Beadle & Company of New York published a set of rules for both Association Football (soccer) and the "Handling game" (Rugby). The first intercollegiate game using rules resembling modern game was played on November 7, 1869 in New Brunswick, NJ between Princeton and Rutgers (Rutgers won 6-4). This game used the London Football Association's 1863 rules which called for, among other things, 25 players, a field 110 meters x 70 meters, a 24 foot wide goal, movement of the ball allowed with all parts of the body (including hands, ball could be batted or held, but not carried or thrown). First team to score 6 points won. Interestingly, this same game is also generally recognized as the first GRIDIRON FOOTBALL game as well.
Soccer was also taken up at this time by Yale, Columbia and Cornell, and reintroduced to Harvard in 1871 in a hybrid form known as the Boston Game, a version which also allowed the throwing and carrying of the ball. At this time, football was still played by a number of different and conflicting rules. In 1873, inspired by the English Football Association's rules unification, Princeton, Yale, Columbia, and Rutgers met in New York to draw up a uniform set of rules based on the London 1863 rules. They established 20 players on a team, a field measuring 400 feet x 250 feet, 25-foot wide goal, 6 goals to win, and a point scored by passing the goal past the goal posts. Carrying the ball was prohibited. Shortly after the first game under these rules, a Yale victory over Princeton, an English team, the Eton Players visited New Haven and played Yale, to whom they lost 1-2, in the first Anglo-American international match. Yale was persuaded to adopt the English custom of 11 players to a side, and subsequently argued for its universal adoption, which was generally achieved by 1880.
Meanwhile, Harvard had become more interested in the Rugby form of the game, and looked for competition against similarly oriented teams. When they happened upon McGill University of Montreal, who had also adopted those rules, the two teams played the first intercollegiate rugby match in 1874. The second of these games was played with an oval ball under English Rugby Association rules, and marks the evolution of soccer into the modern gridiron game. A fateful event which would forever change the fortunes of American soccer took place in 1875 when Yale Harvard and bridged the game gap to play a match under special concessionary rules, which included both goals and tries (later touchdowns), and a 15 man roster. Harvard won 4 goals to none and 4 tries to none. Yale reassessed their position after this humiliation, and decided to adopt the Rugby code. Princeton, (who had watched the game as observers) were impressed enough to follow suit. In 1876, Harvard, Princeton and Columbia formed the Intercollegiate Football Association using Rugby rules. Stevens, Wesleyan and Penn soon followed, and the end of 1876 had signed the death knell signed for collegiate soccer in the US."
