Did he really write all those works? And did he write them by himself?
Most of the time.
The end.[1]
[1] Of the canonical Shakespeare plays according to the First Folio, Wells and Taylor's The Oxford Shakespeare: A Textual Companion (1987, 109-134) considers King Lear, Othello, Hamlet, Two Gentlemen of Verona, Titus Andronicus, Richard III, Romeo and Juliet, King John, Henry IV Part I, Merry Wives of Windsor, Much Ado about Nothing, Henry IV part 2, Henry V, Julius Caesar, Twelfth Night, All’s Well That Ends Well, The Merchant of Venice, A Midsummer Night's Dream Antony and Cleopatra, Coriolanus, The Winter’s Tale, Cymbeline, and The Tempest all to be solo-authored by Shakespeare.
More recent studies informing the New Oxford Shakespeare (ed. Taylor et al, vii-viii) considers Henry VIII or All is True, All’s Well That Ends Well, 1-3 Henry VI, Macbeth, Pericles, Timon of Athens, Titus Andronicus, and Measure for Measure as being problem plays from the canonical Folio Shakespeare corpus. See the complete list of plays the New Oxford Shakespeare (2016) offers on the volume's companion website (library login required to read the material).
More recent studies informing the New Oxford Shakespeare (ed. Taylor et al, vii-viii) considers Henry VIII or All is True, All’s Well That Ends Well, 1-3 Henry VI, Macbeth, Pericles, Timon of Athens, Titus Andronicus, and Measure for Measure as being problem plays from the canonical Folio Shakespeare corpus. See the complete list of plays the New Oxford Shakespeare (2016) offers on the volume's companion website (library login required to read the material).