Childhood Games and topology?

The first case seems like an application of knot theory, and we should expect all possible patterns to be equivalent to the "unknot" (a loop.) Knot theory can indeed be considered a topological field. Any sort of weaving you would do with your hands would be an acceptable deformation of the original loop since you aren't cutting it and you can't force part of the string to pass through another part.

The second one concerns the existence of an Eulerian path in a graph, so I would put it more in graph theory. Graph theory is also very related to topology (in fact if you visit the Eulerian path link, you'll find how it is in the roots of topology.) The existence of four vertices with odd degree precludes an Eulerian path.