That there are no Shakespeare writing samples. The only example of his penmanship we have is his signature on several documents. As with deVere, his signature changes every time. This does not suggest that they were the same person, only that people back then did not develop a single consistent signature or a single consistent spelling of their name.
Basically, writing samples do not tell us anything about Shakespeare or Oxford for that matter.
Edward de Vere and William Shakespeare are two quite different people who lived in more or less the same place at approximately the same time. Even if he wrote everything credited to Shakespeare (and all the evidence there is on the subject, and there is quite a lot, says he didn't and Shakespeare did), he still would have been a different person from William Shakespeare. Edward de Vere is not Shakespeare in the same way that Nelson Mandela is not Barack Obama.
Apart from William Shakespeare? The truth is that everyone without exception in Shakespeare's days and 99% of people now know that Shakespeare wrote his plays. The other one percent have weird notions that someone else wrote Shakespeare's plays. In the 19th century it was Francis Bacon. Nowadays it is more likely to be Edward de Vere. It doesn't matter because it is impossible that Bacon or de Vere or anyone other than Shakespeare wrote them.
Anne Boleyn was Queen Elizabeth's mother. Neither de Vere or Shakespeare had any relation to either woman or to each other.
The real author behind Shakespeare's plays and poems has been subject to a controversy dating back about 150 years. The so-called "Anti-Stratfordians" maintain that Shakespeare, with his provincial background (from Stratford-upon-Avon) could not possibly have collected sufficient knowledge and education to be accountable for his body of work. The most popular candidates as the Bard's ghostwriters are Francis Bacon (1561-1626) and Edward de Vere, the 17th Earl of Oxford (1550-1604); the latter being favored by some on the grounds of Shakespeare's complete lack of productivity during the last decade of his life until 1616.
because he was writing the plays
Edward de Vere was the 17th Earl of Oxford. After his father died he was raised by William Cecil who was the secretary of state at the time. He was a learned man, and spent a lot of time in Italy. Many events and influences in de Vere's life appear in the works of William Shakespeare, and therefore some people believe that de Vere is Shakespeare, but published his work under a name that just happened to be the name of a man who lived in Stratford at the same time (publishing under a different name was very common at the time, but, of course, the vast majority of people believe that William Shakespeare of Stratford is the true author).
The film you are referring to is "Anonymous," directed by Roland Emmerich. The movie explores the theory that Edward de Vere, Earl of Oxford, was the true author of William Shakespeare's plays.
Edward de Vere was the seventeenth Earl of Oxford. He lived between 1550 and 1604. He was a poet and a playwright and is one of the people some think may have written some of the plays ascribed to Shakespeare.
Lady Clara Vere de Vere was created in 1842.
It is possible but unlikely. Shakespeare and de Vere were of totally different social classes and would not have mixed socially. De Vere was an ardent theatre fan and supported two companies (Oxford's Men and the Children of Paul's) as well as the playwright John Lyly in the 1580s before Shakespeare started his career. It is possible that the young Shakespeare may have been connected with the Oxford Company as a hired man, but the patrons of theatre companies had nothing to do with the day-to-day running of the company and so would likely never have had any contact with the hirelings, if indeed Shakespeare was one of them. After about 1590, as Shakespeare's star began to rise, de Vere's steadily diminished. He became a peripheral figure, crippled by crushing debt and ill health. He may have continued his interest in the theatre by attending performances by the Lord Chamberlain's Men, rivals to his own companies, and may have interacted with the actors at such a performance, including Shakespeare. But there is no evidence at all that such an event took place, and because of the social differences involved, it is very unlikely.
Because there is no actual evidence that anyone apart from Shakespeare wrote his plays, the field is pretty well open for wild speculation. There are dozens of people who have been proposed as the author of Shakespeare but the front runners are Edward de Vere the Earl of Oxford, Christopher Marlowe and Francis Bacon. Since neither de Vere or Bacon showed any aptitude for poetic writing, and both de Vere and Marlowe were demonstrably dead when most of Shakespeare's plays were being written, these are pretty sketchy suggestions indeed.
Well, assuming that he did, and the evidence all points the other way, we would have to assume that he wrote because he liked writing. Since there is no evidence that he did write the plays, there is naturally no evidence as to why. He was well off, so it wasn't as if he needed the money (especially since writing plays didn't pay that well unless you were the owner of a playing company).