Feats
Archery
Does the Archery feat work with a Melee weapon that you throw?. No, the Archery feat benefits only attack rolls you make with Ranged weapons. The Thrown Weapon Fighting feat benefits ranged attack rolls made with Melee weapons with the Thrown property.
Charger
If you have the Charger feat, can you use the Charge Attack benefit while mounted?. Yes, but only on a controlled mount.
When a mount moves, you move with it. Therefore, if the mount moves 10 feet in a straight line toward a target, you also move 10 feet in a straight line toward a target. However, you still must immediately take the Attack action after this movement to gain the benefit of Charge Attack. This is feasible on a controlled mount, since a controlled mount moves on your turn.
Conversely, you can't use Charge Attack if you're riding an independent mount, since an independent mount moves on a separate turn, and you're not able to immediately take the Attack action after that movement.
Crossbow Expert
Does Crossbow Expert let you fire a Hand Crossbow and then fire it again as a Bonus Action?. The Crossbow Expert feat allows you to fire a Hand Crossbow multiple times on your turn. But you still must abide by the rules of the Hand Crossbow's Light property, which state that the property's additional attack must be made with a different weapon.
Dual Wielder
Can my character wield two Heavy weapons if they have the Dual Wielder feat?. The Dual Wielder feat allows you to simultaneously draw two weapons that lack the Two-Handed property. All Heavy weapons in the Player's Handbook have the Two-Handed property, so they don't benefit from this feat. (However, if we were to create a Heavy weapon that lacks the Two-Handed property in the future, that weapon could be used with this feat!)
Dueling
Is the Dueling feat intended to support using a Shield?. Yes. A character with the Dueling feat can equip a Shield or hold a Spellcasting Focus or another object that isn't a weapon in their other hand.
Great Weapon Fighting
If you use Great Weapon Fighting with a spell like Divine Smite or Hex, do you get to treat any 1 or 2 you roll for the additional damage as a 3?. The Great Weapon Fighting feat benefits only the damage roll of the weapon used for the attack. For example, if you have this feat and make an attack roll with a Greatsword, you can treat any 1 or 2 you roll on the weapon's 2d6 as a 3. If you cast Divine Smite after hitting with this attack, you can't treat a 1 or 2 rolled for the spell's damage as a 3.
Magic Initiate
If you're a spellcaster, can you pick your own class when you gain the Magic Initiate feat?. Yes. For example, if you're a Wizard and gain the Magic Initiate feat, you can choose Wizard for the feat.
Polearm Master
Can I add my Strength modifier to the damage of the bonus attack that Polearm Master gives me?. Yes. For the feat's bonus attack, you add your ability modifier to the attack roll, as you do whenever you attack with that weapon, and if you hit, you add the same ability modifier to the damage roll, which is normal for weapon damage rolls.
A specific rule, such as the Light property, might break the general rule by telling you not to add your ability modifier to the damage. Polearm Master doesn't have such a rule.
Savage Attacker
Does the benefit of the Savage Attacker feat apply to additional effects like the Rogue's Sneak Attack or the Paladin's Divine Smite?. No. The benefit of Savage Attacker applies only to rolls of the weapon's damage dice, not to any extra damage that a feature or other ability might grant.