Could you post the errors you get?
From my guess, GetRoot().GetNode("AutoLoaded Sibling to current scene").my_method();
is not going to work because unlike GDScript, C# is statically typed, so you must cast the node before using your method, like:
MyNode node = GetRoot().GetNode("AutoLoaded Sibling to current scene") as MyNode;
if(node != null)
node.my_method();
else
// Uh, the node could not be found or is not a MyNode