Can self-driving cars deal with snow, heavy rain, or other weather conditions like these? Can they deal with unusual events, such as ducks on the road?
No, smart cars do not know what to do when surrounded with ducks or flood waters, and it's possible they never will.
As with all machine learning, a computer knows only what it's taught. If an event arises that's unusual, the AI will have less relevant training on how to respond, so its reaction behavior necessarily will be inferior to its routine "standard operating procedure", for which is has been heavily trained. (Of course this is true of humans too.)
Due to liability concerns, when encountering an outlier condition, smart cars will almost certainly be designed by their makers to immediately pull off the road and wait to be explicitly told what to do -- by the human in the car or by communicating with a central command office that exists to disambiguate such confusion and resolve cognitive impasses. When confused, just like a child, a smart car will be designed to seek external assistance -- and is likely to do so indefinitely, I suspect.
That's why, despite Google's recent cars that lack steering wheels, smart cars most certtainly will retain some means of manual control -- be it a wheel and pedals, or at least verbal commands. Given the many forms of weirdness that are possible on the road, it's possible smart cars will never be fully autonomous.
As for bad weather conditions, how well do smart cars currently perform? Nobody outside of a car manufacturer can say for certain. Lidar and radar are superior to the human eye in seeing through fog and snow. But (competent) humans are likely to remain better than a smart car at dynamically learning the limit of adhesion and compensating (since this is a learned skill few smart cars will already know or can learn quickly -- given this car, these tires, this road surface, this angle of road, etc).
Initially smart cars will turn to the human when the going gets rough, ceding control back to them. Once smart cars have driven a few million miles in snow, slush, high wind, floods, and ice, and encountered many ducks, angry mooses, and irate pedestrians, they will have been taught to do more for themselves. Until then, and perhaps for decades yet, I suspect they will turn to mommy and ask for help.
The state of the art AI driving systems utilize stereoscopic/depth cameras for visual perception. Scenarios such as your ducks on the road example would make the system perceive them as obstacles on the road (it doesn't really matter if they are ducks/goats/humans). The base algorithm should be able to circumvent this situation and bring the vehicle to a safe halt avoiding chances of possible disaster. Hence I doubt scenarios such as this would pose much of a problem to today's AI drivers.
Many cars now instead of just cameras, use radars. Snow, heavy rain, and other weather conditions should not affect them at all. Objects like ducks will be detected. The only problem right now is dealing with things like red lights or road signs, as you have to use a camera to see and interpret them.