Nothing at all.
It probably did, but if so it was not reflected in the plays which Shakespeare was writing at the time of Hamnet's death, viz. around 1596. Shakespeare was very close about his feelings and did not tell everyone about them (of if he did, they did not make note of it).
Scholarly guesswork suggests that Shakespeare might possibly have started writing plays in about 1590, when the Queen was about 57. When she died thirteen years later he was writing more than ever, and kept on writing for ten years after her death.
It was called the Elizabethan age after Queen Elizabeth I of England who ruled from 1558 to her death in 1601. Shakespeare spent the first 37 years of his life during Elizabeth's reign and so naturally began his writing then. He continued writing into the Jacobean era, so named after King James I who came to the throne on Elizabeth's death. Shakespeare would have had a hard time writing, say, during the Victorian era which began 221 years after his death.
Shakespeare wrote in poetry, even when he was writing plays.
Most of Shakespeare's writing is in English. There's also some French.
Before Shakespeare began writing plays he was an actor.
Writing was not Shakespeare's only career, and probably not his first.
A quill pen was the writing implement of choice in Shakespeare's day.
No, it is a fictional film about Shakespeare writing Romeo & Juliet.
No, it was his job.
Shakespearean.