What is the best way to take care of a 1940s silk wedding dress that has some rust marks? Can it/should it be cleaned?

Rust marks are unfortunately some of the worst to deal with. Rust degrades the fibres quickly and cleaning requires some chemicals that aren’t suitable for DIY use at home.
If the fabric around the marks is already very brittle, I wouldn’t take the dress to a dry-cleaners. But if the fabric is still stable, it’s worth a try.

With respect to storage, the best solution is to store the dress lying flat. When clothes are kept on hangers, all the weight is on the shoulders and that’s where damage happens first. But if you don’t have space to store it flat, you can make a padded clothes hanger using polyester wadding covered in a cotton fabric. Try to avoid folding the dress too much. Or you can pad the folds with rolls of polyester wadding covered with cotton fabric, so that the folds aren’t too sharp. Otherwise the usual advice applies: no light and as few fluctuations in temperature and humidity as possible.

Kind regards from the textile conservators