I have branches, but no fruit, trunk or leaves. What am I?

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?

John can fit six large chocolate boxes or nine small chocolate boxes into a carton. How many cartons will he require to put sixty-six chocolate boxes into?

Before Mount Everest was discovered, what was the highest mountain in the world?

It's a test that will check your IQ.

Suppose you have one 11 minute hourglass and one 13 minute hourglass.

Can you measure exactly fifteen minutes using them ?

There are people and strange monkeys on this island, and you can not tell who is who (Edit: until you understand what they said - see below). They speak either only the truth or only lies.
Who are the following two guys?
A: B is a lying monkey. I am human.
B: A is telling the truth.

I am the black child of a white father, a wingless bird, flying even to the clouds of heaven. I give birth to tears of mourning in pupils that meet me, even though there is no cause for grief, and at once on my birth, I am dissolved into air. What am I?

In an interview, a boy was asked an unusual question 'How two persons sitting with a table in between them can't see each other?' He was unable to reply. Can you?

There are 2 people near the river and both of them want to get on the other side. The boat can only take one of them. But they got across.
How?

I have a five and a three-gallon jar. Assuming there is an infinite supply of water, how can I measure one-gallon water?