wierd grammar question

Can anyone help me with this grammar issue? There's some things wrong in here but I can't remember WHY.

Bill said, "What's wrong with Peter? He's lost a lot of weight. He looks ill."

Carol answered, "Yes, he's very slender, isn't he?"

and here's another challenge to my language skills.

Bill said, "When is Anne arriving?"

Carol replied, "The next week."

Bill said, "I can't wait to see her."

Carol finally replied, "Me too. She's very nice, isn't she?"

[Tdol]

'Slender' is not really appropriate for weight loss through illness- 'thin' or 'underweight' would work better.

In the second question, say 'next week, not 'the next week'. We say 'next week' to mean the week after the time of talking and 'the next week' when we are talking about a week in the past.