Although the past perfect can certainly be used in this sentence to some effect, there are many other tenses in English which might also add variety and interest.
For instance, who can resist the oft-overlooked but piquant and enticing Conditional perfect progressive? If we combine it with the equally neglected Conditional perfect, an even more expositionally rich tapestry appears; and when we similarly plunge the other verbs into the obscure and gloomy depths of the Past perfect subjunctive, we can envision Ethan drowning in the effluent from the tap, which, trapped in syntax, forever fills his room with water, maybe. We're not sure.
Ethan would have been washing his hands when the doorbell would have rung. Had he run to answer it, he would have been forgetting to turn off the tap.
The temporal relationships in the sentence are quite easily rendered in the form of a graph. The action plots against time as a quantity expressed in dashes, as below.
From his imagined vantage point in the future, Ethan looks back through the present, which is the past from his perspective (❶), and forgets whether or not he can remember not having heard the doorbell in the slightly less distant past (❸). The tap, which may or may not exist in the future, has the potential to be running in the present/past (❷), and the doorbell, which may or may not have been heard, rings at a supposed time (❹) removed by several units from the imagined present (not the current present, but the present as it exists from the perspective of Ethan's as yet unrealized future.Future-descendant unaccusatory temporal displacement †) (❺).
At last Ethan, unable now to decide whether or not he would have been washing his hands when the doorbell would have rung (or even whether he has any hands at all), in desperation attempts to barge into the imperfect future where, no longer able to determine his position in time with any accuracy, he exclaims: "The Unicorn!" and begins to flap his lower lip (❻).
† See "Methods of Digressive Uncertainty in Forced Time-shifted Recursions", CGEL 18th Ed. pp. 21345-7
NOTE: Ethan's eventual dampness cannot be rendered, since it is conditional on an event that would have been happening in the imagined past from a perspective far, far away in the exotic Isles of Langerhans.