[RPG] Does the UA Barbarian Path of the Beast Claw Add the Strength Modifier to Damage

You add your Strength bonus to any melee weapon damage you do unless a rule specifically states otherwise.

As per the rules on what Strength does for you:

You add your Strength modifier to your attack roll and your damage roll when attacking with a melee weapon [...]

In the original printings of the rules, unarmed strikes were a weapon with an entry on the PHB's table of weapons. However, they were removed from this table by errata, and instead the rules for melee attacks state that:

Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons). On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes.

This means that unarmed strikes are not, technically, an "attack with a melee weapon", but they are a "melee weapon attack". The following rules about damage rolls unfortunately also use the "attack with a weapon" wording:

When attacking with a weapon, you add your ability modifier--the same modifier used for the attack roll--to the damage.

Which strictly rules out the unarmed strike. However, Sage Advice rules that when you make a melee weapon attack, by default you add your strength to the attack roll and to damage. This general rule is always in effect unless it is explicitly overruled by something else:

For example, if you make a melee weapon attack with a longsword, you add your Strength modifier to the attack and damage rolls of the attack. [...]

[...] In other words, you follow the general rule until an exception in the game tells you not to.

The Aarakocra's unarmed strike doesn't explicitly include the Strength modifier, but it also doesn't explicitly exclude it. Additionally, other races with modified unarmed strikes published in later material, such as the Tabaxi, do specify that the unarmed strike damage still includes the character's strength modifier, and the implementation of Aarakocra on the official companion site dndbeyond.com includes the strength modifier to damage.

Though the actual letter of the published rules is not as clear as it could be, the most consistent interpretation of the material is that Aarakocra should still include their strength modifier on unarmed strikes, and it is an oversight that such was not included in their description. The Elemental Evil Player's Companion has never had errata published, but one imagines if it did this would have been corrected so as to be unambiguous.

You would calculate the resistance to each damage separately. The modifier should also have a damage type, and you would include it with the dice roll for calculating resistance.

For example, with a melee weapon like a greataxe where you apply your strength to its slashing damage you would calculate \$(1{\rm d}12 + \text{Str Mod})\div2\;\$ (rounding down) slashing damage. And then calculate \$1{\rm d}6 \div 2\$ (also rounding down) fire damage.

So your final formula would be:

$$\left\lfloor \frac{1{\rm d}12 + \text{Str Mod}}{2} \right\rfloor + \left\lfloor \frac{1{\rm d}6}{2} \right\rfloor$$

(Those lines on the sides being the floor/round down symbols.)

RAW would probably be as described in this answer. They reference:

Damage Resistance: If a creature or an object has resistance to a damage type, damage of that type is halved against it.

Damage Vulnerability: If a creature or an object has vulnerability to a damage type, damage of that type is doubled against it.

Resistance and then vulnerability are applied after all other modifiers to damage. For example, a creature has resistance to bludgeoning damage and is hit by an attack that deals 25 bludgeoning damage. The creature is also within a magical aura that reduces all damage by 5. The 25 damage is first reduced by 5 and then halved, so the creature takes 10 damage.

(D&D 5e SRD, Pg 97)