Rain itself won't hurt them, but you do not want them to be wet for any extended period of time. As soon as they are dry, I would advise you to place them in containers which allow them to breathe ( paper or wood) and put them in a dry place. Until the temps drop below 50, I keep mine in baskets in the tractor shed.

George Carver would say no. Here is a great link on storing & growing sweet potatoes. He was one amazing person

http://aggie-horticulture.tamu.edu/plantanswers/recipes/sweetpotatoes.html

Carver used the air drying cure. I am using the USDA method of high heat and HIGH humidity. Good disease frees tubers will store well under either method. I presume that the USDA method would give the highest flavor quality.

This year was a vintage year for sweetpotatoes here. I raised several varieties from Ivis white [tastes just like a white potato], light yellow [O'Henry], Nancy Hall [yellow], and Beauregard [orange].