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I think the 5-14 stat is just what the unidentified ring states. I don't know why it's not the same as the identified +0 stats.
ETA: I took a closer look, it seems like the scaling is fractional. At 10 STR, max dmg goes up by 2.5 per upgrade (rounded), then at 11 STR, it 3.5.
I don't think Brawler's Stance changes the damage at all. You're still attacking with the ring, you just get to keep holding a weapon—for using weapon abilities—unlike the other classes who have to be empty handed.
Other classes can either get the small damage increase with a weapon or unequip the weapon to attack with the ring. The duelist can enter Brawler's Stance to simulate unequipping their weapon.