One of the great advantage of the concept of writing, developed about 5000 years ago, is that you can still read and use the words of someone who is dead. Thus you can still read Jane Austen's novels long after Jane has died. Shakespeare does not have to be alive for you to perform his plays because they were written down.
Shakespeare's plays were not performed in order to make Shakespeare happy. People did not say in 1616, "Thank heavens Shakespeare is dead so we don't have to perform his plays any more." On the contrary, they performed and continue to perform Shakespeare's plays because they are the best plays ever written in the English language. Shakespeare's death did not change that.
The Globe Theatre
With the exception of the period between 1640 and 1660 or so, Shakespeare's plays have been continuously performed in London since about 1590 or so.
Hamlet is the most performed play. As far as I know, none have been prefoomed.
The play Macbeth alludes to the Plot
Shakespeare's plays have regularly been played in London from about 1590 to the present day, with the exception of the years 1642 to 1660.
Shakespeare's plays have been performed continually for most of the last 400 years, and for 350 of them (since 1660) the female parts in the plays have been played by actresses. Starting in the nineteenth century, a number of the male parts were played by actresses too. So you can tell that there have been hundreds of thousands of women who have played in Shakespeare's plays.
Posthumous. This means "after death". All of Shakespeare's poetry which was published at all was first published when he was alive. The same cannot be said of the plays. Some sixteen or seventeen of the plays, including such well-known ones as Macbeth, Julius Caesar, The Tempest and As You Like It were first published in the omnibus volume known as the First Folio in 1623, seven years after Shakespeare's death. One play was first published in 1634 as a separate publication. Of course the plays had been performed many times and were well known before Shakespeare died, but they had not been published.
All of the ones we know about have been performed.
William Shakespeare has been a writer for most of his life. His earliest performances of his plays were on the London stage by 1592.
There is probably no one theatre where all of Shakespeare's plays were performed. You may be thinking of the Globe Theatre. The Globe Theatre was built in 1599 and probably saw all of the plays Shakespeare wrote after that date. But Shakespeare had already been a playwright for seven or eight years before the Globe was built. If any of his plays written before 1599 were played at the Globe they would have to have been revivals. Some of his old plays may have been revived, but all of them? Unlikely.
The First Folio was the first Collected Works edition of Shakespeare's plays; it was the work of his fellow actors John Heminges and Henry Condell, and came out in 1623, some seven years after Shakespeare's death. However, many of the plays had been published earlier. A "first edition" of Romeo and Juliet would be dated 1597.
A theatre. Plays have been primarily performed in theatres since the creation of the genre in Greek times.