There was no "Shakespeare Theatre" in either 1592 or 1613. Shakespeare did not own any part of any theatre until 1599. In 1599, Shakespeare's associates Richard and Cuthbert Burbage, who did own a theatre, wanted to build a new theatre to solve some of their legal problems. They therefore arranged to use parts of their old theatre to build the new one, but they needed some cash to buy supplies. Enter Shakespeare and some other guys who put some money into the Burbages' new theatre in exchange for a cut of the profits. It was this new theatre, the Globe Theatre, which burned down on June 29, 1613 during the performance of one of Shakespeare's plays. To the people back then, the Globe would have been the "Burbage Theatre" since Richard Burbage was more famous than Shakespeare and actually owned a large chunk of it.

the theatre William Shakespeare built. The theatre William Shakespeare built in 1599.

American Shakespeare Theatre was created in 1955.

Shakespeare did not have a theatre in Stratford. There's one there now, The Royal Shakespeare Theatre where the Royal Shakespeare Company plays, but there wasn't one in Shakespeare's day.

Well, Shakespeare's partners were raising money to build a new theatre, for one thing.