Can you find a number that lies one third of the distance between 1/3 and 2/3?

Can you solve the tricky chess rebus puzzle below by ciphering the hidden meaning?

What have one eye but still cannot see?

A man lives on the fifteenth floor of an apartment building. Every morning he takes the elevator down to the lobby and leaves the building. In the evening, he gets into the elevator, and, if there is someone else in the elevator, or if it was raining that day, he goes back to his floor directly. Otherwise, he goes to the tenth floor and walks up five flights of stairs to his apartment. Can you explain why he does this?

It is a six-letter word.
The first four letters are me.
The second and last letters are the same.
The fourth second and last letter is payment.

Who is it?

Thomas has missed an excessive number of days of school, so he must meet with Principal Davis. Mr. Davis asks him "Why on Earth have you missed so many days?" Thomas replies "There just isn't enough time for school. I need 8 hours of sleep a day, which adds up to about 122 days a year. Weekends off is 104 days a year. Summer vacation is about 60 days. If I spend about an hour on each meal, that's 3 hours a day or 45 days a year. I need at least 2 hours of exercise and relaxation time each day to stay physically and mentally fit, adding another 30 days. Add all of that up and you get about 361 days. That only leaves 4 days for school." The principal knows Thomas is full of it, but can't figure out why. Where is Thomas going wrong?

You walk into a room where there are three primates held in their respective cages:
1) A lion who is eating the flesh of a goat.
2) An orangutan who is playing with blocks.
3) A donkey who is sitting idle.

Who is the most intelligent primate in the room?

Different lights make me strange by changing me into different sizes. What am I?

The person who makes it has no need for it. He who purchases it does not use it. The person who does use it does not know they are. What is it?

Rearrange the letters in "new door" to make one word.