Intelligence could be defined as a mental model of the workings of our surroundings, and a capacity for learning. This allows us to discover laws about our environment and to visualize, and then build, tools to alter our environment. A few animals have considerable capacity for learning, like elephants, whose brains grow almost as much as ours during our lifetimes. But most animals have developed sophisticated hardcoded intelligence (instinct) instead of the advanced software and plastic (malleable) brains that we developed. For instance, ants and bees have evolved an amazing sort of collective intelligence where each ant or bee is more like a neuron in one of our brains.
Humans' defining attribute is that we are intelligence specialists. Some animals are speed specialists, or camouflage specialists, or reproduction specialists. Some animals live so short a time that the ability to learn anything would be pointless. We just happen to be the result of another experiment by nature to fill another niche, and we sacrificed a lot to develop intelligence, if you compare our physical abilities with animals; the only other thing I think we have going for us is manual dexterity, so we can make tools, and besides that we pretty much suck in comparison to most animals.
So your question needs to be rephrased. Rather than asking "Why did we develop intelligence?", you should ask, "How is intelligence useful?" Like many specific biological features, such as the eye, our intelligence was something that was useful when it was much more primitive, and as various animals slowly evolved and branched off in different directions, our ancestors were the ones who got the larger brains. Evolution has so far seen a benefit every step along the way the further it pushes our brain, so it keeps doing it and we keep getting smarter, despite the occasionally awful side effects (neurosis, autism, etc.).
Your second question is a classic objection by JWs that I used to buy into, but it really doesn't hold up. Color vision had some uses to us or it would have never developed. Telling the purple poisonous berries from the tasty red ones, for instance. Many animals can see much better than us, but they're much stupider than we are; are they enjoying seeing infrared, light polarization, or the tiniest detail or movement in the landscape? Of course not, they're using that vision to hunt and avoid predators. Our vision is fairly weak compared to some species, but it's useful to us.
Music and the arts serve at least a couple purposes. One is attracting a mate; if singing well works for the birds, why not for humans? To put it more crudely, everyone knows rock stars and actors get lots of action even when they're homely. A second purpose is to alleviate our suffering.
You see, the more consciously intelligent we've become, the more self-aware we've become. Self-awareness has benefits, but the big drawback is that we are more aware of our own suffering and of those around us. It would be easy to get depressed, so our brains have developed counter-measures. Various activities release pleasure chemicals like endorphins and dopamine. Some of these activities may be incidental or accidental and just stuck around; for instance, why is music pleasant to our ears? Maybe just because; but since we can easily produce sounds that we find enjoyable, this adaptation stuck around. The most primitive tribes seem to have music, so clearly this was something that even a hardscrabble life, living off the land, allowed for, as it requires minimal energy to sing or beat a drum, and is a good way to kill time.
Other activities release these chemicals for specific beneficial reasons. Exercising releases endorphins. Positive outcomes, such as building something or discovering how to get some food, release dopamine. These pleasure drug pathways may have evolved by chance, by random variation, but they stuck around because they motivated our ancestors to be active mentally and physically, to explore, build and experiment. This is one specific example of part of what we call "intelligence", just to show you how it all serves a utilitarian purpose. Nature doesn't love us, but if it didn't throw us a bone once in a while (music, good food, sex), we would have all thrown ourselves off a cliff a long time ago.