Your little black dots sound like flea eggs or larvae. If you put them on some white paper and squish them do you get a mark like blood? If so they are definitely flea eggs. You need to treat bedding etc with a spray to get rid of them. The white flaky bits could be skin - my cat gets it occsionally, a bit like dandruff. Did you use a spot on wormer or flea treatment? Reaction to this can cause flaky skin.
She is a lovely little cat by the way and well done for taking her in.
Don't worry about your home becoming infested - fleas aren't too difficult to keep under control.