You might want to rephrase part of your 1st question thus: "… it's (*me) I who got the day wrong"?
I guess it means Jess might not have got or remembered the ‘correct’ day. You’ll have to provide more contexts for anyone to figure out exactly what the ‘day’ here means.
"The thing you have in common" in your 2nd question means Jim told Lulu untruthfully he liked the kind of food Lulu liked and he shared the same hobbies and activities Lulu enjoyed.
Student or Learner