はい, you can. You can use a suitable power charger adaptor to charge the solar lights if needed. The solar lights can be charged like a cellphone if you get the right charger. (only when you have to make the solar lights on and you find no way to get sunlight for days)
はい, most of Judn solar lights can work in rain for 2-3 days. The solar lights can still get visible light from the sky and charged during the daytime, but power will not be fully efficient, can only last about 2-3 rainy days.
はい, the solar lights can be charged on cloudy days but probably not fully charged in a day, it can still use the stored energy from the batteries and light on at night.