Answer 1

A derived class is a class that extends another one. All functionality will be inherited from your derived class.
public class BaseClass {
public string SomeProperty {get;set;}
public void SomeFunction() {
....
}
}
public class DerivedClass : BaseClass {
}
public class Program {
public static void Main(string[] args) {
var derived = new DerivedClass ();
derived.SomeFunction();
}
}

Instead a nested class is a class declared into your class. Based on the visibility modifier you can use it only in your "parent class" if nested class is private or outside if it's public (ParentClass.NestedPublicClass)