Unfortunately I live where its cold and raining 99% of the time. They are trying to build a line from where i live to where I work though so it might be bearable to bike the distance to the station in the rain
The netherlands, a huge cycling country, is also known as a dreary place where it rains more than the sun shines. You just put on a good waterproof outfit and you're good. Cycling heats you up as well, so as long as you have good clothes I would say its doable up to and including freezing temperatures, depending on the road surface.
Hills are why bikes have gears and there are plenty of chilly hilly places with strong bike infrastructure and culture. Not Just Bikes goes off on this subject frequently. Check them out on nebula or youtube for a laugh and some good information.
Please think of putting pressure on your local councils to improve bike infrastructure. GCN recently mentioned that more people are in favor of increasing bike paths (in uk and us studies) but carbrains complain louder, so let yourself be heard!
Summer was unbearably hot, and I decided to commute via a bus with AC instead of a bike, but in september I started using a bike and didn't stop unless there was pouring or there was snow on the ground.
I even in mild rain I just took my bike. First winter season I used a bike at all, not to mention riding in a full face cover, leather or ski gloves with a ski jacket.
I also researched some motorbike rain pants but we're in a decade long drought, so far didn't need them.
One thing tho. If it's raining you better have disk breaks, the clampy ones just slip on wet wheels. I had to re-learn how to stop safely.
Riding into a river isn't a great idea on an acoustic bike either, of course. The main reason is that you don't want to be swept away by the current. The secondary reason is that you'll have to clean and re-pack your bearings etc. (especially axles and bottom bracket) afterwards.
Kinda depend on the manufacturer, but the motor should be waterproof. Throttle, the screen, and the connector though, might not. I'm using a conversion kit on my bike and the equipment is those no name cheapo one, but other than the screen condensation after riding through heavy rain, it's still going strong. You could identify the stuff that need waterproofing and do it yourself though.
I have the opposite problem. If I tried to bike home from work in the summer, I would literally die of a heat stroke. I biked to college and even when I had a 7am class, I would arrive drenched in sweat and have to do a paper towel bath in the restroom.