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.
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.