Can a fighter have two fighting styles active at once? Dueling and Defense, for example. You can benefit from more than one Fighting Style option at a time, as long as they don’t have conflicting requirements, as Dueling and Great Weapon Fighting do.
Does the Archery fighting style work with a melee weapon that you throw? No, the Archery feature benefits ranged weapons. A melee weapon, such as a dagger or handaxe, is still a melee weapon when you make a ranged attack with it.
Is the Dueling fighting style intended to support a shield? Yes. A character with the Dueling option usually pairs a one-handed weapon with a shield, a spellcasting focus, or a free hand.
If you use Great Weapon Fighting with a feature like Divine Smite or a spell like hex, do you get to reroll any 1 or 2 you roll for the extra damage? The Great Weapon Fighting feature—which is shared by fighters and paladins—is meant to benefit only the damage roll of the weapon used with the feature. For example, if you use a greatsword with the feature, you can reroll any 1 or 2 you roll on the weapon’s 2d6. If you’re a paladin and use Divine Smite with the greatsword, Great Weapon Fighting doesn’t let you reroll a 1 or 2 that you roll for the damage of Divine Smite.
Sage Advice Compendium, v2.7 r2021