Mr. Monk Is on the Run (2) - S6-E16
Corrected entry: When Mr. Monk and Natalie escape from the Sheriff at the car wash in Sparks, Nevada, they end up taking the Sheriff's car. They are first seen after this with this car as they are checking the Sheriff's briefcase on the side of the road. Then, the very next scene is with Natalie and the man behind bars at San Quentin (just north of San Francisco and at least a 2 to 3 hour drive from Sparks, Nevada). Then they are seen with Natalie's car at the parade in Riverton, CA (a town between San Francisco and San Quentin). The only way this could have happened is if they drove all the way back to Sparks, Nevada and then changed cars and returned to Riverton, California (at least 5 to 6 hours driving). Since the parade was happening that day, there would not have been enough time to do all of this driving and get to the parade on time.
Mr. Monk Is on the Run (2) - S6-E16
Corrected entry: When Mr. Monk is at the hotel, a teenage girl comes in to change the towels with the puppy Eureka. At the hotel, Eureka is referred to as a "she" by both the teenage girl and Mr. Monk. However, at the end, when Mr. Monk "solves the case" of how the bomb will be set off, he talks about puppy Eureka again. This time, however, Mr. Monk says something to the effect that "he" (referring to Eureka) was wearing a special dog collar instead of saying that "she" was wearing a special dog collar. Not an error that the detail-obsessed Monk would make.
Correction: Monk has an eidetic memory giving him complete recall of everything he sees and hears which would include the dogs correct gender as female and not male.
Correction: We can see that Monks memory is sometimes not perfect, an example would be when he can't remember the exact order of his fears without his list. Monk accidently saying "he" instead of "she" is very likely considering it was the heat of the moment and that he is human.
Correction: They only took Rollins' car to escape and so he couldn't follow them. They had plenty of time to get Natalie's car once they were safe. Mr. Monk was already lying low while Natalie went to San Quentin, and Natalie got a camera put in her purse. So they did several things prior to Natalie going to the jail.