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sender refers to the object that invoked the event that fired the event handler. This is useful if you have many objects using the same event handler.
EventArgs is something of a dummy base class. In and of itself it's more or less useless, but if you derive from it, you can add whatever data you need to pass to your event handlers.
When you implement your own events, use an EventHandler or EventHandler<T> as their type. This guarantees that you'll have exactly these two parameters for all your events (which is a good thing).
Manually cast the sender to the type of your custom control, and then use it to delete or disable etc. Eg, something like this:
private void myCustomControl_Click(object sender, EventArgs e)
{
((MyCustomControl)sender).DoWhatever();
}
The 'sender' is just the object that was actioned (eg clicked).
The event args is subclassed for more complex controls, eg a treeview, so that you can know more details about the event, eg exactly where they clicked.