Shakespeare neither built nor arranged to build any theatres whatsoever. Sorry. That was his friends the Burbages. When the Burbages passed the hat to help build their theatres, Shakespeare was one of the guys who gave them money, but he was not involved with the building in any way.
It is called that because it is a reproduction of the First Globe Theatre which stood near to where Shakespeare's Globe is located from 1599 to 1613, when it burned down. It is called "Shakespeare's" Globe Theatre because at the time that it was built, in 1997, William Shakespeare was far and away the most famous person to have been associated with the First Globe. The name "Shakespeare's Globe" ONLY refers to the theatre built in 1997. The one built in 1599 and the one built in 1614 were both called simply The Globe. At the time they were built, William Shakespeare was not the most famous person associated with them, nor was he the principal owner of them.
1. There is not and never has been a theatre called the Elizabeth Theatre. (There are Queen Elizabeth Theatres in Toronto and Vancouver, but no Elizabeth Theatre), so obviously nobody built a theatre by that name. 2. William Shakespeare did not build theatres. He was not a builder. He was not a contractor. Nor did he hire them. Shakespeare invested in theatres, he did not build them.
The Globe or the Blackfriars. He had shares in both.
The Globe Theater (in London) was Shakespeare's venue.
the globe
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.
None of the theatres Shakespeare was involved in changed their names.
Sure, children are welcome at most performances at Shakespeare's Globe Theatre. They often have school tours there. You are aware that the name "Shakespeare's Globe theatre" can only refer to the theatre built in 1997 and can never refer to the one built in 1599, aren't you?
If by "Shakespeare's Theatre" you mean the Globe Theatre (which was only partly owned by Shakespeare, and which was not built by him), the lumber was salvaged from the earlier playhouse The Theatre. This was an ok name for it, since these buildings were called "playhouses" at the time, not "theatres".
The first Globe Theatre was built with materials and timber taken from the first 'Theatre' in 1599 when William Shakespeare and the Chamberlain's men were forced to leave because the Puritan owner, Giles Allen, refused to re-new the lease. It only took six months to build. Richard and Cuthbert Burbage, who had owned the Theatre were the majority owners of the Globe. The first Globe burned down in 1613 but a second Globe was rebuilt on the same spot and was ready for use in 1614. It was torn down in 1644 to make way for housing. Shakespeare's Globe, a modern replica of the first Globe, was built in 1997.
They built The Globe in 1600, and took over Blackfriars as their indoor winter home in 1608.
When you say "Shakespeare's Globe" you mean the theatre built in 1997. It is not used for musical performance. The Globe Playhouse (which should not be referred to as "Shakespeare's Globe" since this is the name of the modern theatre) which was built in 1599 was not, as far as we know, used for musical performance either.