I don't deal with heavy rain, ice, sleet and snow. Fortunately, we get very little frozen precipitation in NC, and it isn't worth the investment or trouble to install studded tires for such infrequent occurrences. Heavy rain, I just don't like, and thunderstorms are just dangerous. It's messy, obscures my vision (glasses), and gets my bikes filthy. I have called off rides due to wind as well, if gusts are 35 mph and higher, but that doesn't happen very often around here.
Anyway, I wouldn't have any qualms not riding in the weather you are describing. I ride for fitness and enjoyment, not torture.