First question:

Who was president of his country eleven different times, and managed to lose almost half of his country's territory?

The answer is:  Antonio López de Santa Anna, probably the most hated single man in the history of Mexico.  He started his territory-losing career by granting Texas independence.   The Texians (what Americans in Texas called themselves when Texas was part of Mexico) led by Sam Houston, had captured Santa Anna at the Battle of San Jacinto.   He was terrified that they would repay him in kind for his slaughter of prisoners at Goliad and at the Alamo, and wasn´t hard to convince that agreeing to Texas independence was a good idea. 

More information about the life and times of Santa Anna is here.

Second question:

Several Latin American countries have had female presidents, among them Nicaragua and Panama.  Who was the first woman to become president of a Latin American country, and what country?

The answer is Isabel Perón.  The [in]famous Argentine dictator Juan Perón, met Isabel Martínez when he was in exile in Panama in 1955.  At first she was his "personal secretary",  and in 1961 they married.  Juan Perón remained popular among large segments of the Argentine population, and was finally allowed to return to Argentina and run for president in 1973.  He insisted that Isabel be his running mate, and she was duly elected vice president.  When Perón died in 1974, Isabel took over as president.  She was deposed and put under house arrest by the military in 1976, and shortly thereafter the "dirty war" started.  More information about Isabel Perón is here.

Third question:

One Latin American country made a substantial contribution to the United Nations forces in the Korean Conflict.  What country was it?

The answer is Colombia.  Colombia contributed a battalion of infantry, which was equipped and trained by the U.S.  When the Colombians returned home, they kept the U.S. -provided arms and equipment, and of course the training.  They were then used by the government in the never-ending civil war which has gripped Colombia almost without pause since its independence.  Gabriel García Marquez's famous novel Cien Años de Soledad (A Hundred Years of Solitude), for which he won the Nobel Prize is in large part about this civil war (buy C. A. de S. here)  More information about the Colombian Military and its role in politics is here.

Fourth question:

Two Spanish-speaking countries have won the World Cup of soccer (football to everybody but we norteamericanos).  What countries are they?

The answer is Argentina and Uruguay (Argentina twice).  This is a real trivia question, and has no heavy cultural significance.  It is interesting, though, that Uruguay, one of the smallest countries in Latin America is in this elite soccer class; as is the fact that Mexico, with by far the largest population in Latin America, is missing.

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