I would've expected winter time to invalidate options of bike transport, but in city areas where the snow doesn't stay down, it doesn't end up happening as much. You'd of course want to bundle up for the weather, but then it's not so bad. NJB has even talked about how there are some areas of the world that have permafrost under them, and they still prefer bikes. Rain, in my experience, is just miserable either way, so I'd usually prefer the flexibility to go when a downpour has halted rather than keeping a dedicated roof-mobile around to force my way through. That said, buses have been great for rain too, so again, flexibility.

The sweaty mess remark matches with my experience when I was unused to cycling, and when going uphill. The former becomes less true after even just a week or two of experience on the pedals; I didn't even need a ton of acclimation after recovering from a leg injury. The latter may be a symptom of poorer city planning - which prefers bikes treading flatlands for long distances. It shouldn't end up being tiring when you're basically moving your legs in a walking motion at a low pace (traveling for chores is naturally going to be very different from the Tour de France).