Is the growth of a large tree over my mother's grave an indication of her righteousness, or is it a mere coincidence? And is sprinkling water on graves rooted in the Sunnah, or is it an innovation?

There is no text indicating that the growth of trees on a grave is evidence of the virtue of the buried person. In reality, plants have been found at the graves of non-Muslims. As for sprinkling water on the grave, its explanation has been provided previously.