Sleep and other daily activities will revolve around other biological processes, such as eating. Not everyone will sleep at the same time, and that's a good thing.
The example of the arctic is still useful for finding out what rhythms will govern people's daily lives in your world. For animals and people in most parts of our world, their circadian rhythm, their biological clock that tells them when to sleep, is informed by the 24-hour day-night cycle. But animals adapted to living in the polar regions don't sleep for months and then stay awake for months. Caribou (or reindeer) don't seem to have a 24-hour sleep cycle. Instead, they take naps throughout the day. Since they're ruminants, they spend portions of the day grazing and portions of the day digesting, and they nap while they digest.
Humans also like to take naps after eating, especially after big meals. There's apparently multiple theories as to why this is, but suffice it to say that in the absence of a day-night cycle, the people in your world will default to sleeping after they eat. Depending on how long it takes to forage/hunt food (or work for wages to buy things) and then to prepare your meals, a typical day might consist of any number of forage-cook-eat-nap cycles. Times spent cooking and eating will be an excellent opportunity for socialization with whoever else is also awake, as will the brief period of darkness in your world's middle region (since darkness will make work difficult). Of course, that short night can also be spent napping.
But different people have different metabolisms, especially at different ages. Some people need more or less food, and some people need more or less sleep. So not everyone in a community will end up sleeping at the same time. This is not a bug, it's a feature. If everyone sleeps at the same time, you need to assign a watchman to be the one person to stay awake and look out for danger. But if someone in the community is always awake anyway (whether it's because the grandma can't sleep or because the new mom has to nurse the baby every couple hours), you don't need a watchman. Even if your people are advanced enough to have cities and nations so that they don't have to worry about predators or warfare, staggered sleep cycles will still benefit local law enforcement and businesses. There's no struggle to find someone to take the night shift if there's no night.
Welcome, and happy worldbuilding!
