Are the second and third sentences in this paragraph correct?

Emily pulled her iPhone out of her pocket and launched the Safari web browser. Jack saw that she was thinking like him. She searched for "Buffalo Hyatt Regency." In the middle of the results page, they both saw a newspaper article titled Hotel Receptionist Missing.

Perhaps if you just say "Jack realised they were both thinking similarly".

The rest is fine.

I would put " " around Hotel Receptionist Missing.

Yes they are correct

Jack realized they were thinking alike.