Short answers, pulled from the story.
Amerigo Vespucci coined the term "New World" in the spring of 1503 in a letter written in Lisbon to his friend and former patron Lorenzo di Pier Francesco de' Medici. The letter was published in Latin under the title Mundus Novus in 1503-04 and was immediately reprinted across Europe.
Mundus Novus argued that the lands European navigators had reached to the west were not the eastern edges of Asia, as Christopher Columbus had insisted, but an entirely different continent. Vespucci described it as more populous than Europe, Asia, or Africa, and more temperate and pleasant than any region previously known.
No. Columbus continued to insist until his death in 1506 that the lands he discovered were the edges of Asia. In a 1499 letter to the Catholic Monarchs of Spain he suggested the South American landmass was the Biblical terrestrial paradise, not a new continent.
The Waldsemuller map of 1507 was the first to place a completely open sea between Asia and the newly discovered lands with no land bridge connecting them. It famously labeled the southern continent "America". Waldsemuller's own revised map of 1516 walked back this position, merging the Asian landmass into North America and removing the "America" label from South America.
Ferdinand Magellan's circumnavigation, conducted between 1519 and 1522, established that the Pacific Ocean was a single continuous body of water separating Asia from the Americas. Vasco Nunez de Balboa had discovered the Pacific coast of the New World earlier, in 1513.
Common beans, maize, squash, avocado, tomato, and capsicum peppers were domesticated by pre-Columbian peoples in Mesoamerica. The Andean region of South America produced the potato, cassava, peanut, quinoa, and domesticated the alpaca and llama. Other New World crops include cocoa, tobacco, vanilla, rubber, sunflower, and fruits like the guava, papaya, and pineapple.