One may not say Tehillim or other prayers during Pesukei D’zimrah or Chazoras Hashatz.
“…and from my students (I learn) most of all”
I used to open up a sefer (usually a Mishnayos) during Chazaras Hashatz. Then, about two years ago, I found my oldest son reading a book for school during Chazaras HaShatz. When I told him that it was inappropriate to do so, he countered that I also read during the Chazarah. I tried, at first, to rationalize my actions with the fact that what I was reading was actual learning while he was just reading a novel, but, when push came to shove, he was right — what I was doing was wrong too — even if it was in learning. I had to admit to him that he was correct — and I at that point told him that I would no longer learn during Chazaras HaShatz.
As a result, since then, I follow along with the Chazzan during Chazaras HaShatz — and so does he.