Character mistake: It is highly unlikely that the recipient of Mr. Butler's application for the professorship position would not only call him so soon, but ask him to come to her office right away (where she would humiliate him for the things he wrote in his essay on "qualities that make me a good teacher"). Moreover, her failure to listen to Mr. Butler and give due consideration to his assertion that someone else actually wrote those offensive things showed poor judgment, especially since she knew the person who referred Mr. Butler and spoke highly of him. Someone tampering with his application would make more sense than believing that Mr. Butler actually wrote those things that criticized the very college he was applying to and boasted so much about himself. That Mr. Butler was not able to say more to make his point believable is also problematic. He is an English teacher, so should have the vocabulary and mastering of techniques to make a convincing argument, but he failed to defend himself.
Plot hole: It seems highly unlikely that Lucas would be able to "mess with" Mr. Butler's application for a professorship position, or be able to accomplish what he did in such a short amount time. Off-screen, Lucas went into Mr. Butler's briefcase (foolishly left on his desk in the classroom during lunchtime) and (correctly) assumed Mr. Butler's application was on his laptop. Lucas would have to be able to figure out Mr. Butler's password (something that could take forever), find the right file on his laptop, read or scan the application to determine what he could do to mess it up, write a new answer for "qualities that make me a good teacher", and leave the classroom before anyone saw him. Lucas would also have to assume that Mr. Butler would not notice the change when he printed the application, made a photocopy, and double-checked the application before inserting it in the envelope and mailing it.
Other mistake: What Lucas supposedly did to one of his teachers at his previous high school (Paul Garrett at Milton High School in San Francisco) is far-fetched. Paul told Mr. Butler that Lucas also blackmailed him for an 'A' by videotaping him in his house drinking alcohol with an old friend. Lucas hunted down Paul's old friend (how would Lucas know about this friend?) and paid him to get Paul - a recovering alcoholic - to start drinking again (how would Lucas know that his teacher was an alcoholic, eight years sober?). Moreover, it doesn't make sense that someone's old friend would accept money from a teenager to get his eight-years-sober friend to drink. Lucas may be clever and conniving, but he was still a kid - someone educated adults should be able to outsmart... or at least not get duped by him.
Other mistake: Mr. Butler told Principal Fermont that he'd like to talk to Mr. Ward about Lucas, and she gave Mr. Butler Lucas' home address. Mr. Butler went to Lucas' home without notice more than once and even went upstairs into Lucas' bedroom after Mr. Ward - a lawyer - told him to leave. If a teacher wants to talk to a student's parent, the parent would be asked to come to the school for a conference.
Other mistake: The grade report that Lucas received in the mail indicated that they were "Final Grades" (Sixth Cycle). When Detective Speck asked Lucas why he thought Mr. Butler believed he "had it out for him", Lucas responded that he was trying to figure that out but it had been "ever since he gave me an 'F' on my midterm report card." Also, now that instructors in many/most (all?) schools submit grades on-line, students are able to view their grades on-line and hard copies of report cards may no longer be sent to students or parents via USPS. (00:51:15 - 01:03:38)
Continuity mistake: Mr. Butler parks his car close and parallel to his house. The night/morning Lucas slashed his tire, the car was parked away from and perpendicular to the house. It is worth mentioning that instead of Lucas going on the passenger side of the car where he would be hidden between the car and the hedges, he slashed the tire out in the open where he would have been more likely to be seen and caught. (00:27:15 - 00:28:30)
Plot hole: Detective Speck essentially believed everything that Lucas told him about Mr. Butler's "love relationship" with Becca. The detective took the word of a student over that of a teacher and by using what could hardly be called an "investigation." This trusting of what a new high school student said, especially in combination with the teacher's denial, is extremely poor and inexcusable behavior on the part of a detective (who appeared to be old enough to be quite experienced and even retire... years ago). Believing the word of Lucas fed into the plot that Lucas was a sociopath - capable of fooling a seasoned detective and thereby remaining uncaught and not exposed for what he really was - and help build suspense.
Other mistake: The courses on Lucas' grade report are not given numbers or specific titles that would distinguish them from other courses (and levels) within a department. They are listed in general categories: "American History" (a minor distinction), "Calculus", "Chemistry", "English", and "French." A cover page for one of Lucas' assignments shows he was enrolled in "English 301" - still no descriptive title, but at least discernible from "English 101" and "English 201." (00:23:00 - 00:51:54)
Continuity mistake: Lucas received his mid-term grades in the mail BEFORE Mr. Butler actually entered his grade on the computer. Lucas got the mail and took his unopened grade report to his bedroom then the camera shifts to Mr. Butler contemplating what grade to fill in for Lucas. Mr. Butler made the entry (grade not shown) and closed his laptop. The camera returns to Lucas, who opens his mid-term grade report and sees the "F" from Mr. Butler. (00:51:21)
Character mistake: In Principal Fremont's office, Lucas was asked why Mr. Butler thought that he "had it out for him in some way." Lucas responded that he was trying to figure that out himself and added Mr. Butler gave him an "F" even though all of his assignments were "A's." Principal Fremont replied, "Lucas, I don't want you to worry about your grade, okay? I will personally see to it that you get the "A" you deserve." This is not something a principal would say to a student, or say without first checking with his teacher to verify his grades and ascertain if there was a legitimate reason for issuing the "F." (01:03:50)
Continuity mistake: Mr. Butler returned the paper Lucas wrote ("A Defense of Iago"), but it is back in his home office when he uses it to highlight words in it that were also in the paper he believed Lucas substituted in his application to fill a professorship position. (00:35:37)
Continuity mistake: When Lucas and Mr. Butler are playing chess, the timer clocks do not consistently move forward. The timer clock on the left goes from approximately 10.6 to 10.8 to 10.4 to 10.9 and back to 10.4. The timer clock on the right also goes backwards from about 10.95 to 10.8 and then stays at 11 for three moves. (00:12:10 - 00:12:35)
Continuity mistake: The final classroom scene contains some continuity errors. The podium is closer to the board/wall a couple times and is later at an angle instead of being parallel to the wall. The words Lucas wrote on the board - "I have not deserved this" - is in bolder/darker writing during the fight and the word "deserved" is not underlined after Lucas underlined it. There is also no explanation or even hint as to how the detective knew to go to Mr. Butler's class (or the school in general) that night. (00:12:25 - 01:19:18)
Plot hole: Principal Fermont gave Mr. Butler a "solid recommendation" for the professorship position he was applying for and Mr. Butler thought he was going to be hired. However, colleges/universities typically require applicants to hold a PhD in the field of study. Mr. Butler would not be likely to get through the Search Committee's initial screening with his M.A, and would not get hired. [There are far many more people with PhDs than there are openings and tenure-track positions are rare these days.]. (00:33:34)
Character mistake: After Mr. Butler was late getting to his first period English class (due to having a flat tire), he picked up a few markers to write on the white board, but none of the markers had any ink in them. Mr. Butler then told the class, "Let's have a reading day." Public school teachers have daily lesson plans to follow, and a "reading day" would not meet his teaching obligation or fulfill his responsibility. (00:29:30)
Other mistake: Mr. Butler was late getting to his first period English class the morning he had a flat tire (which Lucas slashed). He went directly into his classroom from the door that opens to the outside of the building (which is odd for a school classroom). Typically, teachers must sign in as "present" and have a homeroom period before their first class. Someone would have had to cover Mr. Butler's homeroom and someone (perhaps Principal Fermont) would have gone to his first period class until he arrived. (00:28:35)
Other mistake: Mr. Butler is in Lucas' bedroom, finds a small (4"?) camera, and plays the part where Lucas is setting it up in the classroom to (hopefully) film Becca putting moves on Mr. Butler. Mr. Ward (Lucas' father) is standing in the doorway several feet away and cannot possibly see what is on the small screen, but asks if the girl on the tape is the one who died. (01:07:25)