The red string is still used, but the web of it is seen on the kitchen table at the start of the play, like a quilt the girls are working on. It’s cleared from the stage in the first blackout and brought down from the flys over the kitchen table for the climactic dinner scene. When they pull Henry’s veins out of his leg, they tie it into the rest.
The ceiling here is different than the design of the original production as you described it. It looks like a regular roof of a cabin, but it’s covered in lanterns/glass bulbs and it’s black and shiny. When Henry puts on the extraction mask and then dies, the lantern things glow on and off, and it looks like the roof is covered in souls or fireflies or something.