What does the below rebus riddle means?

When I asked a girl's name she told me her name is the date '07/09/2001'.

I thought for a few seconds and then I got her name. Do you?

It's pretty hard to give up.
If you remove a part of it, you will be left with a bit.
Even if you remove another part, the bit still remains.
Remove one more and it still remains.

What is it?

Can you spot what's wrong with the following sentence?

PETER WAS ANGRY AT
HIMSELF BECAUSE HE
ACCIDENTALLY LEFT THE
THE KEYS TO HIS CAR
AT HIS FRIEND'S HOUSE.

As we know that white starts the game of chess. Can you find the scenario shown in the picture below is possible when all the white pieces are at the original place while the black pawn is not as in the below picture?

What kind of storm is always in a rush?

Using Only Five 5's and any mathematical operator make sum as 37

A person is found murdered in his Chamber. When the crime scene is investigated, the Investigator find out that there is a calendar on which the victim has written a few numbers with his blood. The numbers are 4, 11, 11 and 4.

There are four suspects in the murder, John, Jacob, Anna and Lee. Can you find out who is the murderer?

Can you find a five letter word that is left with only two letters if one is removed?

What is in seasons, seconds, centuries and minutes but not in decades, years or days?