I have heard, and believe it after trying it, that it's a waste of time to attempt to "rehydrate" an old mat. I soaked mine and it was no different afterwards, just wet. It makes sense to me, because the mat is not water soluable or made of water in the first place. It's like thinking you can "hydrate" a piece of plastic, pvc or vinyl.
My guess about the rust is that there is still water or detergent down in some of the cracks in the mat, and when the blade comes in contact with it, it causes the rust. You can wipe down your blade with sewing machine oil after use to help it not to rust. And stop putting your mats in water!