A:would you like to go swimming on Friday, Sep 14, from 2:00 to 5:00 P.M.?
B: I'd like to, but I can't. My body is allergic to water. It causes dermatophytes on my skin. I'm also waterphobic. (not much to correct here. I split the last sentence in two because I think it works better)
C: would you like to watch the horror movie together in the cinema on Sunday, October ninth? (not bad - "with me" works slightly better)
D: I'd like to, but I can't. When I watch horror movies, I can't sleep at night. It causes nightmares and night terror to me.
Not bad... not many corrections here. Well done.
P.S. All these sentences were understood, which is the important thing :)