That battle in the opening was the result of Seifer taunting Squall on for a fight. at the start of the game Quistis does scold Squall for falling for it and in class she reminds the students (i assume sarcastically) to
try and not kill your partner during practice
given that she seems to imply that it's normal for students to spar and there's no restrictions unlike the Training Sector of the Garden (younger students must be with an instructor) it's probably normal but they don't use weapons which can kill (or maybe Junction certain Para-Magic which prevents injury and death which Seifer and Squall did not do) unless an Instructor is around.
given Seifer's "bad boy" image he probably sees the scar as a badge on honor and probably is proud of it. given the one he gave Squall he probably thinks his smaller scar is proof he was the victor in the fight.
As for why Squall even fell for Seifer's taunts, when your walking down the corridor from the Infirmary the choices you have to reply to Quistis both imply Squall was trying to maintain his reputation, one option just have him remain silent (like he always does) while the other he says he couldn't let Seifer get away with it, since we never see that scene i've assumed it went something like this
Seifer: Tonight, you and me and this time for real, no kiddie Para-Magic as a safe guard and no Instructor Trepe to get in our way
Squall: go by yourself, i have better things to do
Raijin: you think your so tough blowing off the boss like that ya know
Seifer: Fuujin, Raijin, leave it, it's not his fault he's pathetic, it just showns he's not fit to be SeeD
obviously i'm assuming that Fuujin and Raijin was there as they probably got the other instructors involved after Squall's Counter Attack, looking out for Seifer as they always do