Short answers, pulled from the story.
The Two Gentlemen of Verona is believed to have been written between 1589 and 1593. Scholars disagree on a precise date: Gary Taylor suggests 1590-1591, Clifford Leech argues for 1592-1593, and Roger Warren proposes somewhere between 1587 and 1591, with 1590-1591 considered most likely.
Many scholars believe it is. The theory was first proposed by Edmond Malone in 1821, and the play has been placed first in several major editions including The Oxford Shakespeare (1986, 2005-2016), The Norton Shakespeare (1997-2008), and The Complete Pelican Shakespeare (2002). The argument rests partly on what critics describe as signs of theatrical inexperience in its construction.
Two friends, Valentine and Proteus, both end up in Milan, where Valentine falls in love with Silvia and Proteus betrays him to the Duke to pursue Silvia himself, having already abandoned his betrothed Julia. Julia follows in disguise as a page called Sebastian. After a confrontation in a forest, Proteus repents, Valentine forgives him, Julia reveals her identity, and both couples are reunited.
The primary source was Jorge de Montemayor's Spanish prose romance Los Siete Libros de la Diana, published in 1559, whose second book follows a woman who disguises herself as a page to pursue the man she loves. Shakespeare also drew on Thomas Elyot's The Boke Named the Governour (1531) for the male-friendship plot, and John Lyly's Euphues (1578) for technique.
Critics are divided. Some, including Stanley Wells, read the line as Valentine literally offering Silvia to Proteus, the man who just attempted to assault her. Others, such as Jeffrey Masten, interpret it as Valentine promising to love Proteus as much as he loves Silvia, reconciling friendship and romantic love rather than choosing between them. Stage productions have handled the moment in radically different ways since the eighteenth century.
The earliest recorded performance was in 1762 at Drury Lane, though this was Benjamin Victor's heavily rewritten adaptation, not Shakespeare's original text. The first known performance of something closer to Shakespeare's version was at Covent Garden in 1784, which closed after a single night. The play had definitely been performed by 1598, since Francis Meres lists it in his Palladis Tamia that year.