When I went to a secular middle school, not THAT long ago, I recall one old-fashioned teacher ALWAYS stated, as soon as he entered the room, “Gentlemen, HATS!” Meaning, that he wanted any of the boys who had hats on, to remove them.
As far as I know, no one ever challenged or reported him for this, even though this was just his personal preference, the school had no particular dress code, and there is certainly not a sin for anyone to wear a hat indoors. I don’t even know if what his religion was, or if he even followed any. It was just his belief that “gentlemen” didn’t wear hats indoors.
Although I guess if he’s still teaching in this day and age, some would be happy to accuse him of stifling the creativity of boys who wanted to wear hats, or some MRA types would accuse of him of being sexist by only demanding the boys remove their hats, or some such.
However, while I don’t think the OP is wrong to issue gentle fraternal correction, I’m not sure if she’s obligated to do so.
As for the second example, unless the teacher is using the school’s resources for her own personal use, I don’t see how using her own cell phone using her own money to access the Internet (as opposed to using the school intranet), unobtrusively is a sin.
As I recall, the legendary Yankee Stadium announcer, Bob Sheppard, always had a book in front of him that he would read during his shifts. Since if, say, a batter was in the box for 10 minutes having some kind of epic struggle with the pitcher, fouling off pitch after pitch, he had nothing to do for those 10 minutes, until the next batter finally came up and it was time to announce his name and number. Not to mention time out between innings and so on.
He didn’t hide it, either. Sometimes the TV booth would show him doing this. But it sure doesn’t seem like George Steinbrenner thought he was “stealing” by doing so!