Should I upgrade from a Canon 350D body or buy lenses first?

Your mom's camera has 8 megapixels. The 18-55mm kit lens is about optimal for a 8 megapixel camera. No need to upgrade that lens without upgrading the camera body.

I'll suggest you to get at least the EF-S 55-250mm IS STM lens. It's much cheaper than the EF 70-300mm IS USM lens. It's also cheaper than a new camera body.

The 70-300mm would be longer at the long end (which is a plus) but at the short end too (which is a minus), while being faster to autofocus, but very much heavier and also more pricey. However, the 70-300mm would support full frame camera bodies too.

Because, if you crop, there's a reason for it. The reason being that you didn't zoom enough. And why would be that the case? It would be the case if you can't zoom enough with the attached lens.

The EF-S 55-250mm IS STM lens allows you to use it in very mild telephoto conditions (55-80mm) that the 80-200mm lens cannot support, so it's more likely that the 55-250mm lens is actually attached to your camera, and also in long telephoto conditions (200-250mm) that the 80-200mm lens can support only by cropping.

The 50mm prime would only be a good idea if you take lots of portait pictures. If taking head-only portraits, consider the 85mm/1.8 lens too, it's more expensive than the 50mm lens, but of higher quality.

Snow on the ground does not matter unless you plan to drop your camera. Snowfall can matter. However, the weather-sealed lenses and cameras are very, very expensive. I'd suggest you to avoid using the camera when it rains, and perhaps purchase a small weatherproofed camera bag where you can put the camera when it begins to rain.