A lot of curtain material (even cotton) isn't meant to be washed. I'd cut a square of the interlining and wash/dry it as a test. When I do these things I write on the fabric itself with a sharpie what the original cut size was (so 12.5 x 12.5" or whatever). I put in a lingerie bag and wash the way I do all my other fabrics which is for me normal cycle with warm water (my washer only offers cold rinse), and then dry on warm. I examine the material as it comes out of the washer on how it holds up, again after drying and re-measure to test for shrinkage and examine the edges for fraying, if any coating is coming off the material, general close inspection.
The easily found recommended batting of choice right now is Warm & Natural (off white) or Warm & Bright (white) or something similar. For me, I find that too heavy and do prefer a poly batting -- and there are all different weights and weaves/processes that give different final results -- but sometimes for the project I am doing W&N is exactly what I want too. There is no single answer.