Some Thoughts. (Because I am too lazy to search for a notebook and write my partial ideas)
From clue 1, we notice that Charlotte is 23 years older than Max. And they share the same last name. It makes me think, Charlotte is related to Max. But, they are not siblings. 23 years is too big a gap. Most likely, Charlotte is the parent of Max. I believe, his mother and the male sex linked with Charlotte is probably a typo.
From clue 2, Paul is Max's father or Alice's girlfriend, which doesn't make a lot of sense, since Paul is a male and Alice is a female. If we go overboard on lateral thinking, then, Paul is possibly a transgender but nothing in the puzzle seems to hint at that. Paul is Max's father seems to be a likely proposition.
Some other thoughts:
I believe, The Lucy clues are just red herring because they lead to precisely nowhere. There are far less clues to make Lucy really mean anything. Last clue says that Alice has known Paul since elementary school. This makes them old childhood friends. Paul is probably Max's father. And Alice is probably Daisy's mother since Daisy and Max are close in age groups and the fact that Paul and Alice are classmates.
Therefore, my vote goes to:
Paul's girlfriend is Alice.
Charlotte is Paul's wife. (Probably divorced, or Paul is having an extra-marital affair).