The PM is a mere servant of the Queen, who is asked by Her Majesty to run the government. So why has the Queen not sacked Mrs. May?

She has no power to do anything, from 1688 the royals have been paid an allowance and as part of getting that they gave their power away to parliament.

When the Queen opens parliament she hands it all over to them. As you stated, they are her servants.

The PM is an elected official. The Queen is careful not to overstep constitutional limits and knows that most of Parliament would have to want May out of office in order for her to even ponder doing that--and Parliament would have to ask the Queen to do so, because her powers come through Parliament. The Queen doesn't like to do the job that Parliament is elected to do.

She cannot even if she wanted to. She has no powers, which is a great relief to us all.

She may dissolve Parliament but, may not decide who can or cannot be an M.P or who their choice of leader is.

The Queen CAN dissolve Parliament but HM has NO ability to dismiss Mrs. May. Even if there was need to!! She wasn't elected by the people but by the Tories after Cameron walked/resigned, she was elected by her Party.

Tradition is that the incoming leader of the elected Party, goes to the Queen for permission to form the Government as chosen by the people. It would be very unusual for HM to have denied that permission.

In any case, WHY would May need to be sacked other than via a motion of no confidence, personally or in the Party - which is going to happen soon as Corbyn has tabled such a Motion of No Confidence - in the Government. May has been working her butt off, stuck between a rock (the EU) and a hard place (the UK Parliament) for a very long time. People may find her attempts to sort out this situation not good enough, but who else is there!! Corney is full of criticism, but hasn't ever come forward with some solution!!! And ditto for all the others in Westminster.

If ever there was a case for pulling together, regardless of Party, it's RIGHT NOW. I hope Cameron is please with how he's left the country.

She's frightened of her.

The Prime Minister is not employed by the Queen and is not selected by her. She merely appoints the PM and it is up to the majority party to propose or dismiss their leader according to their own rules.