Factual error: The patient is diagnosed with legionnaire's disease, caught from the AC unit above their desk. The legionella bacteria causing the disease is found in warm water (commonly dehumidifiers, industrial AC cooling towers, hot water tanks) - the key point being that a water source is required. The problem with this is that the AC unit above the patient's desk is a window unit. These sorts of AC units don't use water as a coolant - you can't catch legionnaire's from them.
Continuity mistake: When Baby Shoes fires the gun at Joe, the first camera shot shows he's holding the gun with both hands. However, in the next camera shot he's only holding the gun in one hand. (00:01:55)
Character mistake: House tells Foreman to search the cop's home, car and workplace. He says that he will focus on the precinct. But the cop's workplace is the precinct.
Suggested correction: They don't use water as a coolant, but they can collect water that condensed from the cold inside the unit.
LorgSkyegon
Which still doesn't allow for legionella to breed, not least the limited amount of collected water drains away (i.e. not like in water tanks and cooling towers). Healthcare professionals have consistently confirmed that window AC units aren't legionella risks for this reason. If you check the CDC website, it explicitly states that window AC units aren't potential legionella sources.