Must bathroom linoleum be sealed and water tight?
I would say that it absolutely should be sealed well all the way around the bathroom, and so that water cannot get underneath fixtures.

However, even on a floor that had no sealing at all I would not expect this kind of damage simply from mopping and minor spills. Do (did) you actually observe water being wicked out from the edge? Could it be that the toilet or another fixture was leaking and getting under the linoleum?