ETA - just saw the pics, HOLY COW what a quilt!! Lucky you!!
I just received an old quilt from my husbands Aunt and it was badly stained. I don't know how old it is, guessing maybe 1930's, but I first tried the buttermilk/lemon soak in the bathtub overnight (didn't work very well). Then I tried oxy & dawn and the water became instantly brown. After a few short soakings, rinsing and gentle squishing, most of the staining came out and the quilt looks awesome! I spun it gently in my frontloader and layed it on a sheet in my entryway to dry.
I would wash in a front loading machine, warm water, gentle cycle and use a product called Synthrapol. If there are any dyes that come out in the washing cycle the Synthrapol will suspend them in the water and they will be eliminated when the wash water is drained. For the rinse cycle I would suggest cold water. I would recommend drying in the dryer on low heat until completely dry and not putting it out in the sun to dry. I would not want to risk the possibility of the sun bleaching out any colors over time using this drying method. However, if you can dry it in a consistently shaded area, then that would be absolutely fine.