What are the problems that developed in the "island society"?
Good question. It might be useful to divide the problems into several categories.
There are problems of survival. This refers to the way that the boys had to build shelters, learn to find food, learn to hunt, start fires, etc.
Related to these are problems due to isolation. They have only limited choices of what to eat. They have only a finite number of kids.
There are problems related to rescue. Because they have no radio, etc., they have to try to keep the fire going.
This leads to the biggest category: social or cultural problems. The boys have problems early on simply forming a new society. They have problems due to lack of social structures and lack of knowledge. (There are no adults, and no rules they have to follow.) The stresses of isolation and survival combine with the effort of forming a society to create new social problems. Some of the boys don't see things the same way, or don't want to follow the rules. That's why Jack rebels. Some of them go a bit crazy (Simon), and that leads to a larger social problem of murder. There's a civil war going on when the adults finally arrive.
The first problem is that they had to live in a nomadic lifestyle, which means they had to find food for themselves, make tents for shelter, create fire for warmth, just to fulfill their needs and basic necessities. They are also in a quest for survival and isolated from the outside world where they have to fend for themselves against the beast and horrible apparitions lurking inside.
The next problem is about the "fake" society in the island, which there is no apparent rules, nothing to govern themselves and they do not have the capacity, the knowledge to differentiate between right and wrong, appearance against reality, so they can't really conform to the "make-believe" societal rules. They have to govern themselves and even degenerate to such lowly acts like brief moments of murder and deception, that had completely change their perspective of life and change them to complete absurdity.