A syke is a small stream. Who knew?
A sheugh is a ditch, perhaps filled with water, that makes a field boundary.

So the first verse says that the birch doesn't grow in water. Well, very few trees do. Very few trees have bark that you can easily get off to make hats. Try to do it with an oak tree - won't work. I think the main point of the reference is to show that the sons have been living rough and in poverty. The magic references don't seem very convincing.

By the way, don't take bark off birch trees. It's bad for them.