Will the rainy day or cloudy/rainy day be a problem for the solar outdoor lights?

It will not be a problem at all.
INLUX SOLAR’s solar outdoor lights are all IP66-IP67 waterproof.  You don’t need to worry about it.
Moreover, when designing solar outdoor light scheme, we will refer to the climate data of the installation site and design a certain amount of solar panel and battery allowance according to the possible continuous rainy days. So you can keep the solar lights on every night, even in rainy weather.