Generators, what for?

Coming up with more questions. Since I was going to observe the behaviour of an elliptic curve for varied generators, I started to re-read about generators and cyclic group.

And here are the questions:

  1. How do generators affects the curve? And its security?
  2. We need a generator because it generates all elements in the group. Means that we can use all the distinct elements. Is that correct?
  3. Why do we use cyclic groups?
  4. What is the difference between cyclic groups and cyclic subgroups? When to use?