1. Cornell Big Red - (picture the color, not the gum) There's no official mascot. The unofficial one is a bear.
2. I had no idea and had to look it up. Pretty random, really. In hindsight, they could have come up with better.
The nickname "Big Red" for Cornell teams originated in 1905. The late Romeyn Berry '04, then a recent Cornell graduate, was writing the lyrics for a new football song. Since Cornell had no nickname at the time, Berry simply referred to Cornell as the "big red team" and it caught on.
The first live Cornell bear mascot appeared in 1915 during Cornell's undefeated and national championship football season. Since that time, there were three others.
The last edition - Touchdown IV - was never allowed on Schoellkopf Field despite strong publicity campaigns by the Cornell Daily Sun and undergraduate groups.
The bear was invited to Cleveland by the Cornell alumni in the city, and was then shipped in a dog cage, to Columbus, Ohio for the Ohio State-Cornell game. At this point, the Animal Protective League stepped in and decreed that the bear was to be let loose in the wilds of Western Pennsylvania.