I’m studying sculpture at school and can’t find any info that tells me why most statues and busts have upturned mouths. Why were they sculpted that way?
You’re overdoing this question.
First, the “upturned mouths” are not Roman. I think you’re talking about conventional facial expressions on some Archaic Greek Kouroi and Korai. See the site below.
Second, over the centuries (and that includes Roman portrait busts and even the Fayum encaustic mummy portraits), portraits have been intended to be solemn and dignified. Until the very late 18th century, people who spent money on having a portrait painted intended to be seen in a dignified social role, not simply laughing or hanging about for the fun of it. These were documents of how the patrons wished to be seen by posterity, not necessarily how they really were.
18th-century patrons relaxed about this to some extent, but even today, prominent, distinguished people who commission formal portraits do not expect to be depicted laughing or smiling. See the series of portraits of U.S. Presidents and First Ladies at the White House.