1. The day before yesterday, is two days after Wednesday. What day is today and why?
Just answer what day it is today. First sentence is not a question. My guess to what they are after, anyways.2. Even if every single person in the world dies, human race will continue to exist. How come?
Every person dies, but not at the same time. No riddle here..3. Who's cat thinks outside the box when it exists?
Not sure about this one. Sounds like a Schrodinger's cat ref. as mentioned, but you are right, there are two grammar mistakes.4. What is the airspeed of an unladen swallow?
11 m/s This one you can google. But yes, it's a quote from Monty Python and the Holy Grail.5. Why are prime numbers awesome?
Just bring up an example, like the cicadas.
6. After learning about the greatest number ever defined and used in mathematics (show us some Google skills), how would you "write" every single digit of this number?
You are right, as far as I know. It's the Graham's number.
A definition of a number, is just as much a number, as a series of digits.
Both are explanations, describing the number, using a known system, and other numbers. Just write the definition.
But these are my answers, the point is for you to come up with your own after all =) Good luck.