overdrive
Not use their high beams and drive dramatically lower speeds.
Turn on your headlights and your wipers drive a bit slower than when in good weather
The overdrive doesn't care if it's raining or not, all it cares about is the speed. If you're driving fast enough for the overdrive to work, then you should use the overdrive, regardless of weather. If you isn't, you shouldn't, regardless of weather. Often it's sensible to lower speeds in poor weather conditions.
Overdrive use depends on speed, not weather. Can you drive at overdrive speeds, use overdrive. If you can't, don't. Often it's sensible to reduce speed in poor driving conditions.
low beam
Yes, when driving in foggy conditions you should use your headlights which allows oncoming drivers to see you at the earliest possible moment. However, for safety, you should not use your highbeams [bright] as the fog will reflect back much of the light, reducing your ability to see into [through] the fog.
Use low-beam headlights. When visibility is restricted, a driver's natural tendency is to activate the high-beam headlights. When driving in fog, this further impairs visibility because the high-beam illumination reflects off of the fog and back at your vehicle.
Yes, dramatically.
it does not
Over drive button shuts overdrive off - should be used when towing, when engine braking is desired for descending a small hill, when in sustained city traffic not exceeding 45 mph, when driving in inclimate weather snow etc.
Driving
False. You may confuse the opposite way driver of your real location.