When Harvey is waiting for the judge outside the court house, Harvey hands the judge a complaint (paper) that is on top of an envelope but when the scene shot changes toward the judge the complaint paper is no longer visible.
Mike receives a reply from the patent office on the same day he makes the initial filing; this is impossible as the backlog of unexamined patents extends at least three years.
Harvey tells Mike to file for a patent. While any attorney can file a patent infringement claim, only attorneys who have passed the federal patent bar (the "Examination for Registration to Practice in Patent Cases Before the United States Patent and Trademark Office") can file for patents. And only attorneys that have extensive scientific or technical backgrounds are permitted to take the exam. Even if successfully pretending to be a lawyer, Mike would not be able to file for a patent.
When the parties attend the injunction hearing, the seal on the wall behind the judge (Currie Graham) indicates they are in a state courthouse. State courts have no jurisdiction over patent disputes, which are governed exclusively under Federal Law.