Yes they do have a soccer team.
my fifa11-edit teams-international selections
The United States played its first international soccer game in 1884 and they have had an international team from then on.
At the current, the manager of the Irish international soccer team is Giovanni Trapattoni.
no because Puerto Rico doesn't have an international team and Mexico is rivals with the US
The team started soccer as a whole in 1893, but it joined FIFA/International Soccer in 1912. Sources: FIFA.com
The name of Kenya's team is simply the name of their country - Kenya. This is true of any international professional soccer team.