If you are using other managed objects that are using unmanaged resources, it is not your responsibility to ensure those are finalized. Your responsibility is to call Dispose on those objects when Dispose is called on your object, and it stops there.
If your class doesn't use any scarce resources, I fail to see why you would make your class implement IDisposable. You should only do so if you're:
Yes, the code that uses your code must call the Dispose method of your object. And yes, the code that uses your object can use using
as you've shown.
(2 again?) It is likely that the WebClient uses either unmanaged resources, or other managed resources that implement IDisposable. The exact reason, however, is not important. What is important is that it implements IDisposable, and so it falls on you to act upon that knowledge by disposing of the object when you're done with it, even if it turns out WebClient uses no other resources at all.