I’m japanese native speaker. My major is not linguistics.
I write just my opinion. I think your recognition of the two “気が利かない” and “空気が読めない” is correct.
another example of 空気が読めない人 is someone who speak loud in silent space like museums.
気が利かない and 気が回らない have almost the same meaning and usage. I think most Japanese people don’t know the differences between them.
気が利かない、気が回らない、空気が読めない contains bad nuance in itself. But 気が付かない is often used in daily conversations. So there are a lot of possibilities that person can’t notice. If he/she can’t notice others help or potential choice to help someone, 気が付かない has very similar meaning to 気が利かない. But 気が付かない can also use for someone who are 鈍感(unresponsive, dull). For example , someone doesn’t notice even he/she is called by another person.
気が知れない人 is used when others can’t understand the person’s (人) thought because his/she(人) often think things in different way from common sense.
This used often in bad nuance because in Japan, different from others is bad thing.
空気が読めない人、気が利かない人、気が回らない人 is person who can’t understand his/her around.
気が知れない人 is person who others can’t understand his/her actions or behaviors.
So the subject to “can’t understand” is different.
気が付かない、気が知れない is I think often in bad nuance(気が知れない is 90% bad word), but sometimes it’s not. You need to care about the context.
気が利かない人、空気が読めない人、気が回らない人 気が知れない人 can be understood immediately
is I think natural. because above three example has only bad meaning. 彼は気が知れない is vague but I can understand that he is unusual person whether in bad way or not.
is not natural for me. This sentence need to add information about what he don’t notice because there too many interpretations.