Short answers, pulled from the story.
The first opera was Dafne, composed by Jacopo Peri in 1597, with a libretto by Ottavio Rinuccini based on the myth of Apollo and Daphne. Only the libretto and small fragments of the music survive. The first complete surviving opera is Euridice, also by Peri and Rinuccini, from 1600.
The Florentine Camerata was a society of scholars and musicians sponsored by Count Giovanni de' Bardi in late 16th-century Florence, formed to study drama and music, especially Ancient Greek theater. Its members, including Vincenzo Galilei and Girolamo Mei, concluded that Greek performances used individual sung voices rather than polyphony, and set out to recreate that model. This led directly to the first operas by Jacopo Peri and Giulio Caccini.
Monteverdi's La favola d'Orfeo of 1607 introduced the named aria as a structural unit, an orchestral opening he called a symphony, and the ritornello, an instrumental passage repeated between acts. He also specified distinct voice types for principal roles: Orpheus as tenor, Eurydice as soprano, Charon as bass. His L'incoronazione di Poppea of 1642 was among the first operas built on historical rather than mythological subject matter.
Francesca Caccini, daughter of Giulio Caccini and herself a singer and composer, is credited as the first woman to compose an opera. Her work was La liberazione di Ruggiero dall'isola d'Alcina, composed in 1625.
The Teatro San Cassiano, founded in Venice in 1637 in a palace belonging to the Tron family, is considered the first public opera house in the world. It was established by Domenico Mazzocchi, and the first opera performed there was L'Andromeda by Francesco Mannelli. The theater was demolished in 1812.
La púrpura de la rosa, composed by Tomás de Torrejón y Velasco in 1659 with a text by Pedro Calderón de la Barca, is the first opera both composed and performed in the Americas. It was premiered at the Viceroyal Palace of Lima.