First off if you want to use an excellent dog shampoo and conditioner, then try the #1All Systems. I use it an find it to be the best, and I've tried many dog shampoos on foster Yorkies over the many years.
As for training. If your dog will not go out in the rain, then I suggest on rainy days to put him in an expen with a wee-wee pad at one end and attach it to the front of his wire crate. Put in a kong and a nylabone and attach the water dish to the side of the pen. Pet stores sell this type of dish as well as Petedge.
www.petedge.com/exercisepens
I don't take mine out in the rain at all. When it rains I just put a couple of wee-wee pads down. I'm not fond of having them soaking wet several times a day.