If a three-month dark period was a normal, recurring, part of a 15-month calendar, and plants had evolved to be bio-luminescent, could they survive? Would only some survive and, if so, would it be enough to keep a large human population stable (or would there be a type of culling for the lower-class people)? Would this change if many animals were also bio-luminescent?
Can a plant survive a 3 month period of no luminosity (or at least in amount that does not provide enough light to sustain any photosynthesis)? Yes, it's called WINTER, in case of polar night the tolerated period without light can be even longer.
Being bioluminesent would be quite expensive from evolutionary perspective. There would be an evolutionary pressure to drop this folly, unless there is some strong benefit for plant.
By occasion - low sunlight generally tend to mean lower temperature. That may be problem of its own.
Animals can adapt, unless such change would happen overnight, then we would face another mass extinction event, especially in tropical regions.
Humans (in A.D. 2019)? In long run it would be easily adaptable, we could squeeze much more production per hectare if the prices were higher. The tricky part would be surviving first year. I don't think that there would be any culling of low classes per se. There would be culling of livestock and food rationing, and if that's not enough, then 1st world would halt (either officially or simply by overbidding) its food exports towards third world. Even in pessimistic scenario a low class person from first country would become much slimmer, while watching on TV how people in Africa resort to cannibalism.
Yes, just like plants on Earth create stores of energy in their roots to survive winter, your plants could develop a similar strategy of survival. It is also very likely that the leaves of said plants will fall off during this darkness period as leaves take energy to maintain and would not be returning energy without light.
As for human survival, the plants would not be doing much growing without an incoming source of energy, so harvesting from these plants during the darkness period would be difficult. However, humans are more than capable of preserving food ahead of time to be used throughout this period.