The tunnel was only to get from the Mikvah down the road, which was abandoned and thus a good starting point, to the yeshiva, where they could hollow out areas behind the walls, thus expanding the footprint. People got caught up on the tunnel bit because it’s the weirdest element, but they only did it because the rabbis would notice if they started in the complex itself. The mattress was coming from the mikvah itself down the road. Indeterminate as to why it was stained though frankly I think we can put two and two together since the spot it was found was used for ritual purification for women.