It was all about putting on plays for the public to make money for the players and writers.
It depends what you mean by "Shakespeare's theatre". Do you mean the theatre which was built in 1996 and is called Shakespeare's Globe Theatre? Or do you mean the theatre company which he joined, the Lord Chamberlain's Men, which was founded in 1594? Or do you mean the tradition of Elizabethan theatre of which Shakespeare was a part, which started in about 1560? It could be any one.
the theatre William Shakespeare built. The theatre William Shakespeare built in 1599.
American Shakespeare Theatre was created in 1955.
Depends which Swan Theatre you're talking about. There was a Swan theatre in Shakespeare's day, but it's likely that it never saw a Shakespeare play, because it was operated by the competition. On the other hand the Swan theatre at Stratford has seen lots of Shakespeare plays, probably all of them.
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.
Shakespeare bought share in a theatre group where worked for five years. The name of the theatre is Globe theatre.
It's a Theatre someone has decided to name after Shakespeare. The most famous of these, although neither is strictly speaking called "the Shakespeare Theatre", are Shakespeare's Globe Theatre, built in 1995 in Southwark, London, and the Shakespeare Memorial Theatre, originally built in 1932 and substantially renovated since, in Stratford.
There is no one theatre which can be identified as "Shakespeare's theatre". Shakespeare performed in a number of different theatres, although most particularly in the two he owned a share in. In any case, the main reasons for theatre closures applied to all the theatres whether they had something to do with Shakespeare or not. There were two of these: all London theatres were periodically closed for a relatively short time for health reasons, and all London theatres were permanently closed in 1642 by law.
If there was a theatre called "William Shakespeare Theatre", you will have to be a little more specific. Was there such a theatre built in Akron, Ohio in the 1930s? Or in Calcutta in the 1890s? If the theatre you are talking about is "Shakespeare's Globe Theatre", it is still standing, having been built in 1997. If the theatre you are talking about is the Blackfriars Theatre, in which Shakespeare acted and held a small share, it was demolished in 1655. If the theatre you are talking about is the First Globe Theatre, in which Shakespeare also acted and held a small share, it burned down on June 29, 1613. If the theatre you are talking about is the Second Globe Theatre, which was built to replace the first one in 1614, and which might have had nothing to do with Shakespeare, it was torn down in 1644.
William Shakespeare was a minor shareholder in The Globe Theatre. He, as well as the other owners, was a member of the theatre company that played there, The Lord Chamberlain's Men later called the King's Men. Since Shakespeare was the company playwright, all of his plays were played by that company and often at that theatre (but not always)
The Globe Theatre
It depends what you mean by "Shakespear's Theatre". Do you mean one of the two theatres he invested in in his lifetime? They have the same names they always did. Do you mean a theatre famous for its Shakespeare performances? There are lots of those all over the world. Do you mean a theatre with the name "Shakespear's Theatre"? If so, I don't know if there is such a theatre, but if there is it must be called Shakespear's Theatre, obviously.