Lots and lots. Some plays like A Midsummer Night's Dream, are full of them. Even plays that don't rhyme often have rhymes at the end of a scene or long speech to emphasize the fact that this is an ending, such as "The play's the thing/ Wherein I'll catch the conscience of the king." (Hamlet)
Yes. A Midsummer Night's Dream is a good example of a comedy full of couplets.
Yes, and no. Shakespeare uses many different styles of language, such as blank verse, rhyming couplets and ordinary "vernacular" language.
couplets
Shakespeare wrote 38 plays.
Various kinds, but mostly blank verse (unrhymed iambic pentameter). He often rhymed them in couplets as well.
The Elizabethan theater was used for many of Shakespeare's plays.
Most of Shakespeare's poems are sonnets, but there are also long narrative poems written in couplets.
There were exactly 63 plays that shakespeare wrote by himself
All the known Shakespeare plays are printed in books.
Shakespeare wrote many plays but these are just two of them. Macbeth and Hamlet.
At least 37 plays.
They are not really unique. There are many many plays by Shakespeare's contemporaries, many of which are better than the worst of Shakespeare's plays. Shakespeare was just one of a large number of people writing plays. But what history has shown is that Shakespeare, when he was on his game, was a far better playwright than any of his contemporaries and anyone else before or since. He was not unique, but he was and is the best.