Can we add variables and properties in interfaces in C#.NET?

Interfaces are contracts to be fulfilled by implementing classes. Hence they can consist of public methods, properties and events (indexers are permitted too).

Variables in Interfaces - NO. Can you elaborate on why you need them? You can have variables in Base classes though.
Properties in Interfaces - Yes, since they are paired methods under the hood.
Members of an interface are implicitly public. You cannot specify access modifiers explicitly

public interface ISampleInterface
{
    // method declaration
    bool CheckSomething(object o);

    // event declaration
    event EventHandler ShapeChanged;

    // Property declaration:
    string Name
    {
        get;
        set;
    }
}