I ride in the rain as long as I have good visibility and the surfaces aren't too slippery. My horse is barefoot and trimmed every 5 to 7 weeks so she is fairly sure-footed. We don't often get snow here so I can't comment.
In wet conditions, I avoid slopes, any gait above the trot and sharp corners. My mare is spooky at the best of times so I like to take her out in different weather conditions including thunder storms as long as it's not a violent storm. She will spook at more things as she's already stressed but that's not a bad thing when approached properly. She has to learn from me that the world isn't a scary place and that I won't lead her into a harmful situation. We are still building trust...she's only been mine for six months so we have work still to be done but are getting there with regular ground work and riding.