What is this cutting instrument Tomoko has, and is she supposed to have it at school?
That's a box cutter, more commonly known as a utility knife. I'm unsure exactly how common they are in Japanese schools, but they're certainly not prohibited: off the top of my head, Charlotte and Grisaia no Kajitsu are two other anime that depict box cutter-wielding schoolgirls (albeit in more sinister contexts).

In real-world Japan, there was an infamous incident in 2004 where an 11-year-old schoolgirl slashed a classmate to death using a box cutter. There were discussions afterwards about whether to lower the age of criminal responsibility, but I can find no record of any discussions about banning box cutters from schools, which suggests to me that there is a cultural element to their continued use, when other countries have banned them over far less.

(As an aside, I went to school in the UK and I do recall using box cutters during secondary school craft lessons, but they would have been rounded up and locked away with the other supplies after each lesson, and I'm pretty sure you would have been in massive trouble if you were caught walking around with one outside of those lessons.)