The thing that I saw in this "riddle" was that there was nothing that either implicated anyone as the "Top Suspect" or exonerated any one of them.
English teacher - The Karaoke was not clearly stated as being done with anyone else there. As far as we know this was just some letting loose some steam during a break. So if there are people there this is either a valid alibi or the other witnesses to the alibi are lying to cover her lie. If there was no one there then this is either her way of taking a break or is a lie about her activities.
No smoking gun here.
Science teacher says - Dying one's hair is easy to prove or disprove. Hair dyes are very obvious by their smell and/or the boldness of the hair color. So if the person did dye their hair, that is not an alibi, but is an account of their activities. However since depending on how long it takes to dye the hair (45 minutes to an hour for long hair on average, shorter for short styles) there could still be time to dye one's hair and still commit the murder.
It's a wash. Suspect perhaps but not the obvious murder.
Math teacher says - This is the one where I was called a stupid idiot at best (and the worst I will not mention here). Everyone who says "First day of school, what grades?" is missing out on the point that I remember the "Well kids, it's the first day back from Summer Vacation, let's see how much you remember from last year" pop quizzes that some teachers nail you with on day one. Even more so with standardized testing (don't get me started on that fiasco). So the teacher could be telling the truth and could be doing grades from the earlier periods or she could be lying to cover her butt.
Still a person of interest, but not worthy of hauling off in handcuffs right there and then.
P.E. teacher says - This is the least likely in my eyes. After all there is a classroom of witnesses to the alibi. However this does not clear the person since they were in a room with students. This person could have killed the student and intimidated the rest to lie to cover the murder. Wouldn't be the first time.
So you have someone who has an alibi but it may or may not be true.
Guard says - Again someone with an alibi, but like the Gym and Math Teachers above, this may not be true. The witnesses could be lying.
No smoking gun.
You can argue that anyone of them could be the killer based on the parameters of the riddle. You could also argue that none of them did it using the same parameters.
This riddle lacks enough information to answer it one way or another.
And I was called an idiot for not jumping on the "IT'S THE MATH TEACHER!!! THEY'RE THE GUILTY BUGGER!" bandwagon.