1. What is the pointed part of a cone called?

The top point of a cone is called the vertex.

2. Is the vertex of a cone always directly above the centre of the base?

Not necessarily!
If it is, the cone is called a right cone. 
We usually assume a cone is a right cone unless told otherwise.

3. How can you find the slant height of a cone?

If you are given the radius of the base and the height of the cone, you can use trigonometry / Pythagoras' Theorem to find the slant height.