Can you solve the below tricky visual equation?

A high school has a strange principal. On the first day, he has his students perform an odd opening day ceremony:

There are one thousand lockers and one thousand students in the school. The principal asks the first student to go to every locker and open it. Then he has the second student go to every second locker and close it. The third goes to every third locker and, if it is closed, he opens it, and if it is open, he closes it. The fourth student does this to every fourth locker, and so on. After the process is completed with the thousandth student, how many lockers are open?

A wife and her husband were driving in their car on the highway. All of a sudden, they ran out of gas. So the husband said to the wife, 'Now, you stay here. I will go down the highway to the nearest gas station, and it will be about 1 hour. Just listen to the radio and read some books, and remember to lock all the windows and doors. I will leave the keys with you.' So, off the husband went, and the wife first locked ALL the windows and doors. Next, she turned on the radio, and this is what she heard on the news report:
'THERE IS A MURDERER ON THE LOOSE. HE WAS LAST SEEN ON THE HIGHWAY, WEARING ALL BLACK, ABOUT 5 FOOT 11 INCHES. PLEASE, BE AWARE AND CONTACT THE POLICE IMMEDIATELY.' The wife got very scared. She turned off the radio and double-checked the locked doors. Then she saw the murderer, only a couple of feet away from the car. An hour later, the husband returned to his car. Inside the car, his wife was DEAD, she had been murdered. All the windows were still locked, and the doors. No windows were broken, and the car was in PERFECT condition. No scratches or anything, it was the same way as when he left it. How did the murderer kill the wife?

I speak without a mouth, and hear without ears. I have no-body, but I come alive with the wind. What am I?

A guy in his thirties is found dead in a forest. The shocking thing about the body is that he is wearing his swimming trunks snorkel and a facemask.

For your information, the nearest lake is some 10 miles away and the sea is hundreds of miles away.

How did he die then?

A man was convicted of a minor offence in Akbar court. Akbar decided to give him a chance. He asked him to give a statement. If the statement is true, he will be killed by lions and if it is false, he will be killed by trampling of wild elephants.

The convicted person requested help from Birbal and since the crime was not a big one, Birbal decided to help him. Whatever Birbal suggested impressed Kabir and he let the convicted person go.

What did Birbal suggest to the person?

A murder has been committed in a house. You are a detective and have to find out the murderer.

You investigate by asking three questions to each of the six suspects. Out of those six suspects, four are liars. It is not necessary that they speak everything a lie. But in their answers, there must be at least one lie. One of the six is the murderer.

There are eight rooms in the house in which the murder has been committed: Kitchen, Living Room, Bathroom, Garage, Basement, 3 Bedrooms.

At the time of the murder, only the murderer was present in the killing room. Any number of people can be present in any of the other rooms at the same time.

Can you identify the murderer and the four liars? Also, can you find out who was in which room?

The responses of all the suspects are mentioned below.

Joseph:
Peter was in the 2nd bedroom.
So was I.
David was in the bathroom.

Mandy:
I agree with Joseph that David was in the bathroom and Peter was in the 2nd bedroom.
But I think that Joseph was in the living room, OH MY GOD!

Peter:
Mandy was in the kitchen with Christopher.
But I was in the bathroom.

David:
I still say Peter was in the 2nd bedroom and Jennifer was in the bathroom.
Joseph was in the 1st bedroom.

Jennifer:
Peter was in the bathroom with Christopher.
And Mandy was in the kitchen.

Christopher:
David was in the kitchen.
And I was in the 2nd bedroom with Peter.

PS: The corpse was found in the Living Room.

Give it food and it will live; give it water and it will die.

I am first found in caves, now prolific online; I am a depiction, a drawing, a symbol, or sign. I will convey whichever mood you could wish; or for that matter, a fist, flask, or fish. What am I?

I sleep during the day and fly at night, but I have no feathers to aid my flight. What am I?