This is exactly why we need test characters: making them shows where the holes are. For the most part I've been saying "that could work" etc. since initially this was a very undeveloped idea, and I wanted (and still do want) to incorporate peoples' input, to make sure it's a setting other people will want to play in.
Spirits are not invisible by default (though perhaps some have the ability) but they are rather good at hiding, due to the ability to phase into objects (and other unorthodox things, like hiding in peoples' shadows). As for their shape, it sort of emerges in conjunction with their identity. As they start to decide who they are, they also decide what they look like, though the identity forming process (as in humans) is not an entirely voluntary one. Their appearance might change over time if their personality does, and most are able to change their appearance to some degree at will (though in either case this malleability varies greatly from spirit to spirit.)
As for any kind of society of theirs, well, spirits are sort of living contradictions. On one hand, they all emerge from unified formless chaos, on the other they've all become specific individuals, and just as there might be a human with a chaotic personality, there can be spirits with orderly ones. Among many other reasons, they don't have a society per se because they are relatively few (hundreds of thousands at the fewest) and spread out across the globe. However, they do have some general understandings and guidelines amongst themselves, based on mutual self interest. In their general experience, drawing attention from humans leads to trouble for them, and so often a spirit will try to keep other spirits in its area from being too conspicuous or otherwise messing things up. Some spirits have even decided to make it their business to go around trying to boss around other spirits. Of course, some spirits simply do not care what other spirits or humans do, some don't care about the consequences of their actions. And a rare few have even gone their whole existences without meeting another spirit.
Their numbers are much lower than those of humans mainly because new spirits come into the world at a far lower rate than new humans do. Though they can't age to death, they can still be killed (through means that still need to be worked out, though humans and other spirits would probably be the most common cause of death), and some of them just immigrate back to the spirit world. There are probably a lot more spirits there than in the physical world. I'm still not sure for how long the connections between the two worlds have existed.