Why do demons care about mortals? Why do they want to possess them and mess with the world?

It doesn't really gain them anything. Nothing is accomplished and it doesn't benefit them in any way. But the big difference that makes the mortal world so much more interesting than the demon realm is that everything done in the physical world is permanent. When you kill something, it remains dead. When you create something, it remains until it is destroyed. In the demon realm, everything is possible, everything exist as potential. But it also means nothing is real. Nothing a demon does really makes a difference. The demons constantly fight with each other because it's their nature. But with nothing ever staying dead and an infinite amount of time, everything happens eventually.
In the physical world, everything you do makes a difference. Everything has consequences that are permanent and when things happen one way, all the infinite numbers of other possibilities become impossible. Actions have meaning. They make a difference. To a demon who discovers this and finds a way to enter the physical world, this is endlessly fascinating. The meanings demons give to things are completely arbitrary and not at all based on the needs and feelings of mortal beings. And unfortunately for the creatures of the physical world, most demons don't care the least for how other creatures feel about the consequences of their actions. In fact, the entire concept about fighting with others about what possible future becomes the only reality is a great source of joy for many of them. What's the fun of shaping the future if you go along with the same things that other beings want? Fighting against other sentient beings who have opposing plans for the future makes things much more interesting. It's not that demons enjoy causing pain and suffering, they just enjoy being contrary.
Which, somewhat ironically, sometimes results in demons taking the side of mortals and helping them fight the other demons. Not exactly for the benefit of the mortals, but to have an arbitrary reason to fight with their fellow demons. When demons are helpful and show mercy to mortals, it's because they arbitrarily decided to pick this form of ethics, simply for the challenge. Having spend an eternity in the demon realm, where they will go back to once this temporary excursion into the physical world comes to end end, demons have incredible patience. They are in no hurry and the only threat to them is that someone might destroy their current body and banish them. Though evne that's not too much of a problem as they will have an infinite amount of time to find another one and return for another round. But eventually a demon might get bored with what it's doing and completely arbitrarily decided to try out something else with the physical world. Then there is very little that could stop them from turning against their mortal allies. However, it also means that there is a very slim chance that one can convince a demon to abandon its destructive ways and instead try to accomplish something that many mortals would consider "good". Unlikely to happen, but with demons everything is a possibility.