I think your original assumption may be incorrect. ALMOST can by matched with many types of words, not just adjectives. Actually, you could say almost people, but it would be unusual: "Those chimpanzees are so intelligent they are almost people." "They are growing up so fast . . . they are almost adults." "I'm nearly finished med school. I'm almost a doctor."

Almost midnight just means there is a small amount of time before midnight, or you can say it for noon. If it is 11:45 AM you can say it is almost noon. However, there is no subdivision of people. People are either people or they are not, there is nothing that needs to be qualified so therefore it wouldn't be used. So it can be used when there is something that would be close to it, but not in a case where that wouldn't make sense. You could make a joke of something like "it is almost food" if it tastes bad. But that would not technically be possible, it is either edible or it is not. Hope that helps.

This is an interesting question. I can't tell you why we say it this way, but I can give you some other examples. We sometimes use this format with temperature also. For example: "It is almost 85 degrees in here!" or "I was freezing while walking home. It is almost zero degrees outside!" We also use this with days and months. "It is almost June." "It is almost Friday." Grammar rules are sometimes different from the way people talk. Good luck!