The Spanish-American War was named for the two combatants, Spain and America.
It was fought in an effort to make Cuba independent from Spain.
There were basically two different "types" of battles: the one in Cuba (Battle at San Juan Hill) and the one in the Philippines (Battle of Manila Bay)
It began with the Naval victory at Manila Bay which destroyed the Spanish Fleet stationed there and effectively isolated the Philippines. They then invaded Cuba and established their forward...
No, it was limited to the Caribbean and the Pacific Ocean.