Water Stones: At my last approval, I was given to understand that this was treated as a single school, since the two applications were relatively low in power, and the base spell had no major applications except to "create" water that had previously existed. If the new application (Running Water) is too powerful to include in the school, I can back down the power or lose Running Water completely.
Emet: Essentially, the golem has a steel-strength weapon made of the same stuff he is. If it's separated from him, it falls to dust, same as any other part.
Umbral Anchor: Would this work better if it ties the stone to its target in such a way that it follows with them when they teleport or plane shift? As in, if the target teleports while a Shadow Stone is affecting them, the Stone teleports with them and continues to affect them for the remaining duration. I could also drop the plane shifting part, if necessary.
I know it's more powerful than is strictly allowed overall, but I wasn't sure how powerful to make these new abilities, since I hadn't seen many/any like them before. I'll work with you as needed to get the new abilities in line.