Short answers, pulled from the story.
Shakespeare's canon consists of approximately 39 dramatic works, though the exact number remains a matter of scholarly debate. The uncertainty stems from questions about attribution, collaboration, and plays that may have been lost.
The First Folio is a collection of 36 of Shakespeare's plays published posthumously in 1623 by his fellow actors Henry Condell and John Heminges. It is the source of the traditional division of the plays into tragedies, comedies, and histories, and it preserved approximately half the plays that had never appeared in print during Shakespeare's lifetime.
Problem plays are works that resist the standard categories of tragedy, comedy, and history established by the First Folio. Modern criticism coined the term for plays that either elude easy classification or appear to deliberately break generic conventions; examples from the canon include Measure for Measure, The Merchant of Venice, and Troilus and Cressida.
Yes, collaboration was common in the period. Documented or strongly supported co-authors include John Fletcher on The Two Noble Kinsmen and Henry VIII, Thomas Kyd on Edward III (assessed as 60 percent Kyd and 40 percent Shakespeare), Thomas Middleton on Timon of Athens and possibly Macbeth, and George Peele on Titus Andronicus.
Two plays are definitively lost. Love's Labour's Won is listed by the writer Francis Meres and in a bookseller's inventory but no copy survives. Cardenio, co-written with John Fletcher, may survive only as Lewis Theobald's 1727 adaptation Double Falshood, though modern scholarship has not confirmed that any original text underlies it.
In 1642, England's Parliament banned all public stage performances, including Shakespeare's plays, on the grounds that the theatre promoted "lascivious mirth and levity." The ban lasted through the Interregnum until 1660, when theatre resumed in limited form following the death of Oliver Cromwell.