Why Did Mrs Kearns Write Run?

Mrs. Kearns writes "run" on the paper she slips to Teddy because she knows he has an opportunity to escape while they're doing the whole role play experiment. It's also why she sounds "coached" about what to tell Teddy - she has been.

Why does Mrs Kearns glass of water?

After some chit-chat about Rachel and a man named Dr. Sheehan, Mrs. Kearns appears to drink an invisible glass of water, with the follow-up shot showing a physical glass but no water inside. ... From Teddy's perspective, he blocks out the water because it reminds him of a traumatic experience.

What did the note mean in Shutter Island?

And also, they are barred from searching Ward C and told that the lighthouse on the island has already been searched. The only clue is a note, which reads: "The law of 4; I am 47; They are 80; + you are 3; But who is 67?". A code that Teddy believes points to a 67th patient, when there are only 66.