Technically yes. But there just isn't enough potential energy in the water falling on a typical house.

For example, US average yearly rainfall of 76cm, on a 10m*15m roof, at height of two stories at 5 meters, would give you 1.5kWh of energy per year. That's enough to run the appliances in your home for an hour.

You might be in a more rainy location, or have a larger home, but the end result is still insignificantly small.

It has a more fundamental problem because there's just not that much rain falling on most roofs outside of places with monsoon seasons. Because of this even with cheap efficient energy storage there's just not very much there to harvest.

Yes, and the maintenance issues of using that waste flow.