The Lord Chamberlain's Men performed at the court of Elizabeth I and also at that of James I and his consort Queen Anne.
When King James I started paying Shakespeare's bills Shakespeare's Acting troupe changed its name to The King's Men.
Troupe
The acting company to which Shakespeare belonged for most of his working life was called The Lord Chamberlain's Men from 1594-1603, and The King's Men after that. Shakespeare was not the troupe's leader, but rather a partner.
Queen Elizabeth was queen when Shakespeare was born and continued to be the queen until she died, when Shakespeare was 39.
The Theatre, and the Curtain in Shoreditch, north of the Thames
The Globe Theater is the only playhouse built in 1599 in which Shakespeare's plays were performed.
If you are talking about England, the country where Shakespeare was born, Elizabeth the 1st was the Queen
troupes worked with a troupe of traveling actors
Shakespeare and his fellow actors appeared before the Queen and performed many times. There is no reason to believe that the Queen paid particular attention to Shakespeare.
Shakespeare's troupe, originally known as the Lord Chamberlain's Men, changed its name to the King's Men in 1603 after King James I ascended to the throne and became their patron. This change reflected the support and patronage they received from the monarchy, allowing them to flourish in the theatrical world of the time.
Queen Elizabeth
Queen Elizabeth I arranged to have her cousin Queen Mary of Scotland killed in 1578, when Shakespeare was fourteen. This is the only queen who was killed at this time.