The attached photo was taken in Idaho Falls, Idaho in the winter of 1978-1979. The white stuff is snow and yes, the Model A does handle well in snow and ice when equipped with a set of chains, even without added weight in the back to increase traction (provided reasonable care is taken while driving) although a little more weight in the back certainly wouldn't hurt.
These snow chains (with cleated cross links) were made from a set of chains no longer needed for our 1969 Cadillac Coupe de Ville and also served well during a short (3-4 day) snow storm when we lived on a hill in east Bremerton, Washington (c 1980, if memory serves) and I was using the A for a daily driver.
The chains sit unused since moving back to California in 1982 (I also haven't had to unfreeze a door lock since then, thank goodness).
Additionally, driving in the snow (or cold weather in general) is made marginally more comfortable by a manifold heater (Otwell Health Heater, Model 14 in my case) which provides warm air to the driver's side but will "fry" the passenger's feet after everything reaches operating temperature (so I guess - on the average - it's not too bad...).
As to rain, the best recommendation has already been stated: Rain-X. Don't leave home without it.
[As an aside, it is surprising what a little snow will do. In Newport News, Virginia, during the winter of 1965-66, we received about a 1/4" of snow that lasted less than a day. We had just transferred from Idaho and were accustomed to driving in snow; not so many Tidewater drivers. The "light dusting" of snow basically brought the area to a standstill with drivers skidding through intersections willy-nilly and other such events. Brought new meaning to "defensive driving."]
