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In school perhaps. But Shakespeare was not forced to write after he left school. Most actors were not also playwrights. But since Shakespeare could write and was very good at it, and was paid to do it, why not?
Shakespeare's very first school was probably a "dame school" run by women in town, in which he would have learned his letters, and basic penmanship. Since there are very few records of these schools, we can only guess what the children were expected to write. Passages from the Bible, perhaps.
No.
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Henry VI.
A quill pen. That's what they used then.
Shakespeare's play is based on Plutarch's Lives of the Greeks and Romans, which Shakespeare almost certainly took at school.
Shakespeare's first printed work is thought to have been the poem Venus and Adonis, printed in 1593.
He did not write any novel.
A teacher whose name we do not know.
For Shakespeare, it took William from 1598 to 1601 to write the play.
There is not enough evidence to make an educated guess.