Which number should replace the question mark?

A detective visits his friend John just to discover that John's neighbour Dr Jacob was murdered. Dr Jacob, who is a mathematician wrote the murderer's name in a code word "Hcqtfz". There were five suspects
1. Ajit
2. Pankaj
3. Ganpat
4. Deenu
5. Hariya

The Detective was instantly able to find the murderer.
Can you?

What is harder to catch the faster you run?

Can you find the next number in the below sequence
3 , 7 , 10 , 11 , 12 , 17 ?

Twice ten are six of us,

Six are but three of us,

Nine are but four of us;

What can we possibly be?

Would you know more of us?

Twelve are but six of us,

Five are but four, do you see?

What are we?

Can you find out the remainder when 3^300 is divided by 5?

I travel all around the world but always stay in the corner. What am I?

We are a group of eight brothers. We consider the weak, yet we protect the king in every battle. If we move ahead, we never turn back. Who are we?

A Doctor always sneezes before the rain. Doctor sneezes, Will it rain?

You are a thief and you are being punished for your crime. People have tied your head down on a tree with a rope that has been anchored in the ground. A candle is burning below the rope which is slowly burning it away. Just below your head, a Lion has been left loose and is waiting for you to drop down on the ground so he can have you as his lunch.

You have to survive the scenario. How will you do it?