If they are quilt tops without batting or backing, and really need to be washed before finishing, then the safest way to wash them is to first baste the top to a foundation layer of muslin and then hand wash. Unfinished tops can be ruined by washing if they are not first basted to a foundation, since all the seams can fray and the pieces can become distorted.
If these tops were stored in a cardboard box, it's possible that some of the staining may be due to acid from the box eating into the fabric. All wood products (including tissue paper) are acidic unless specially treated to make them archival safe.
Probably the safest product to clean them with is Retro Clean:
http://www.retroclean.com/q&a.html
I have not used this product myself, but several people posted about it here.
If your aunt does not know how to baste a top to a foundation, she might need to mail the tops to you so you can do that for her. I would not attempt to wash a top any other way.
For cut-out pieces, I would soak them in Retro Clean and then starch them before trying to put them together.