Yes
Both were written in Early Modern English, the same language.
They were written at approximately the same time.
Shakespeare belonged to the Church of England as did everyone else in England and so he used the same Bible as everyone else. Later in his life this would be the Bible commissioned by King James.
James I was King of England when Shakespeare died, on the 23rd April 1616. It is widely accepted that this was Shakespeares birthdate and deathdate, though there are no official records to confirm this. He was 52 years old.
Probably not. It has been said that Shakespeare worked his name into the King James Version of the bible. At the time Psalm 46 was translated, Shakespeare was 46 years old. The forty-sixth word in the King James Version of Psalm 46 is "shake," while the word that is forty-sixth from the end is "spear." While this seems like too many appearances of 46 to be coincidential, it may very well be. At least three translations prior to the KJV use the words "shake" and "spear." The placement of the words, however, varies. It may be that someone translating the KJV doctored the translation to put each half of Shakespeare's name forty-six words from each end of the psalm, but the words themselves were probably not selected for that reason. It is noteworthy that the New King James retains the same phenomenon.
Elizabeth I and James I, but not at the same time. Elizabeth up to 1603 and James after.
Oliver Cromwell was leader but not the king
There was no King of England in 1620 but there was a King of Great Britain. Namely James VI and I (same person).
elizabeth 1 and James 1
Shakespeare helped found Lord Hunsdon's Men. They became the Lord Chamberlain's Men when he received that title, and the King's Men when James I became their patron, but it was the same company throughout. They merely changed their brand name.
The same things as in 1602 and 1604: wrote plays and acted. And, like everyone else in England in 1603, he mourned the death of Queen Elizabeth and celebrated the accession of King James I.
Quite a few monarchs lived at the same time as Shakespeare including King Philip II of Spain, the Shogun Tokugawa Ieyasu of Japan, Sultan Murad III of the Ottoman Empire, and the Holy Roman Emperor Rudolf II. In England, Queen Elizabeth I and King James I ruled successively.