What is both possible and impossible at the same time?

Can you solve the rebus below?

You can find some missing letters in the picture. By placing two particular letters in the spaces, you can form a nine lettered word beginning from one of the corners and going clockwise direction to the middle. Can you find out the letters and the word?

What can you hold in your right hand, but never in your left hand?

Count the number of triangles in the below picture?

One of the leprechaun is missing if you see the two pictures attached to this question. Where did he go? When he comes back, do you know where has he been?

Many have heard it, but nobody has ever seen it, and it will not speak back until spoken to. What is it?

One fine day, an intellectual man came to the emperor's court with the aim of testing Birbal's wittiness. In order to do this, he challenged Birbal to answer his question and hence prove that he was as intelligent and witty as he was said to be.
He asked Birbal, Do you want me to ask one difficult question or a hundred easy ones?
Since both Akbar and Birbal had had a tough day and were eager to leave, Birbal hastily told the intellectual to ask him a single difficult question.
Intellectual: OK. Tell me what came first into the world, the egg or the chicken?
Of course, the chicken, Birbal replied with a smile.
This time with a note of victory in his voice, the intellectual asked Birbal, How will you demonstrate that?
What did Birbal say?

You are playing as white and given four rooks to checkmate the black king in four moves with the following rules 1. You can place one rook every move and ensure the black king should be in check position.2. After four moves the black king should be in the checkmate position.

Without looking at a calendar, within a minute name, a boy's name using 5 consecutive 1st letters of 5 consecutive months