This is not balanced with the other races in the PHB.
Assuming that these are two different sub races ...
- You are starting with a feat.
- You are starting at size large: no PHB race is large. (Not even Goliath)
Why this may matter:
- Weapon damage changes. You get 1d4 bonus when you go from medium to large (see potion of growth, enlargement spell). There is a reason all player playable races are medium sized. The PHB doesn't explain how damage goes up for a Large sized PC because No PC is sized Large. It's a default.
- Monsters get additional damage dice if large. (This should not apply to you, though).
- Being large also allows you to Grapple and Shove Huge sized opponents. It also means that small sized monsters can't grapple or shove you (Kobolds, Goblins). That becomes a non-trivial benefit as you fight things larger than large at higher levels. You can ask my half-orc champion versus giants, cyclops, etc, on that score. Without a potion/spell, his shield master shove does not function.
- Unarmored AC is overly generous, given that class isn't yet given.
Natural weapons at 1d6 + str ... significantly more than PHB races. (Though similar to Tabaxi from Volo's ... 1d4)
The base class benefits of climb, cold resistance, etc fit well enough on their own for the base race. You added a further two skills in the searching for perfection. (Equivalent to half elf skill boost).
Your Proposed bonuses:
For Eir: Dex +2. With two feats. Similar to vHuman.
For Hiojuth: Str and Con + 1. Similar to vHuman.
Compare to variant Human: +1 to two abilities and one feat, and one added skill.
Skills: Your "nimbles claws" as thieves tools is an equivalent to a vhuman added skill, but it isn't a skill. Using it as a bonus action begins to approach a feat. Add to that unarmored defense and you are out of balance.
Add Athletics Proficiency, which makes for three additional skills for the Hiojuth sub-race, isn't too far out of whack. You then added the half-orc racial unique ability for Hiojuth on top of that. And then unarmored defense ...
Natural Armor: natural armor (6 when un armored, ac = 10 + nat armor + dex mod)(maybe changing that to con mod?)
Natural armor as presented is equivalent to roughly plate armor with no shield: 10 + 6 + either Dex or Con mod. (Depends in how high your dexterity is at roll up, to which you have already added 2). If your starting Dex is 16, you have AC 19 without armor. A monk with Wis 16 and Dex 16 has a 16 AC. If you picked a monk, and you have a decent wisdom (14) and 16 dex (Eir), your unarmored AC is 20.
No, this is not balanced.
What to fix?
Size: Medium. Like all other races in PHB and Volo's. (Except for the small ones ...)
Base race: Good! But change Natural Weapons to 1d4 + bonus
Sub races: Lose the feats. Keep the rest.
Eir: Make nimble claws "equivalent to thieves tools" but drop the bonus action.
Unarmored AC: lose it, or trade one of the other base class attributes for a +3 bonus.
Notes.
The unarmored attribute is usually a class ability (Draconic Bloodline sorcerer, Barbarian, Monk, although a few monster races like Tortles or Lizardfolk have something like this).
An argument for "lose it" comes from @Ethan, who made a point in comments about how the game treats Armor Class, and bounded accuracy: the unarmored defense pushes up against a 5E design principle. There are almost no plain AC bonuses. Typically, the game presents an alternate calculation like similar to mage armor. "While not wearing armor, your AC is 12(or 13) + dex mod" would fit the design model more closely. (See the Sorcerer example in the PHB).
Its viability depends on external factors
It is mostly viable, actually the only viable ranged Barbarian build in my opinion1.
The external factors are:
- party composition
- available space
- encounter type (if it is a single boss, or many smaller monsters)
- enemy type (if it uses attacks)
When I played this build, it was quite useful when everything was optimal.
However, its usability decreased significantly with one of these factors missing.
Party composition
As mentioned in the comments of this answer, this tactic is only good if your party members can keep the target of your Ancestral Protectors (Target) away from you.
Either they have to physically hinder its movement, or discourage it with threats of Opportunity Attacks.
You will want to have at least half of the party members to be in melee.
When the Shield Master fighter was not present at our encounters, my Barbarian worked a lot worse.
Available space
If you fight in a dungeon, sometimes you just can't get far enough from the target while maintaining line of sight.
Encounter type
If there are lots of enemies, you might get far enough away from the Target, but his comrades still might be adjacent to you, causing disadvantage on your attacks. You have to hit your target to trigger Ancestral Protectors, so this can be a big problem for you.
Enemy type
If the enemy does not use attacks, only breath weapons or spells, Ancestral Protectors become irrelevant. Unfortunately the resistance is only against his attacks.
Conclusion
It is not viable for dungeon crawling with a party of 4 Sorcerers, but with the average party in the average encounter you will do more than fine.
Recommendation
Use a Hand Crossbow instead of a Longbow and multiclass to Rogue instead of Fighter.
Hand Crossbow
With the Crossbow Expert feat you can attack without disadvantage even if you are adjacent to enemies, negating the worst consequences of a tight space and many smaller enemies.
When you already have the feat, Hand Crossbows give you the most DPR .
Rogue
2 levels of Rogue make Disengage a bonus action, very valuable in tight spots.
Also a Rogue Multiclass provides better DPR increase than a Fighter2, and this area is where you are very much behind a usual Barbarian, as you can't use Reckless Attack and the damage increase during the rage.
1 Being at range gives you extra survivability, by making it harder for others to target and hit you. Ranged characters come away from most fights undamaged in experience, so resistance is mostly wasted on them, and no other significant Barbarian features are usable at range. Other ranged Barbarian's usefullness in combat is below that of a Ranger, and they are famously weak.
2 A single level of Fighter might still be a good idea for Archery fighting style.
Best Answer
Most of these questions involve the use of ability modifiers. You can find the table that tells you the modifier for any given ability score on page 13.
It's a bit late, but off the top of my head...
Initiative is described on page 177 as a "Dexterity Check." This means that you roll a D20 and add your Dexterity modifier.
With a 12 Dexterity, your modifier is +1.
For purposes of the character sheet, you just write the +1 into the initiative box.
Each skill has an associated ability. To make a skill check, if you are proficient, you roll D20 + ability modifier + proficiency bonus. If you are not proficient, you roll D20 + ability modifier.
So your medicine skill (wisdom) would be D20 +3 +2 . On your sheet, you would write +5 in the blank. Your stealth skill (dexterity) would be D20 + 1 . It would be written as +1.
Skill checks are described on page 174.
Saving throws are similar. If you are proficient, write down the ability modifier for that ability plus your proficiency bonus. If you are not proficient, simply write down your ability modifier.
Passive checks are described on page 175 as:
For you, this would be 10 + your wisdom modifier plus your proficiency bonus.
You write down your attacks. For each weapon you have, write down the weapon name, your attack bonus (generally strength modifier plus proficiency modifier for melee and dex modifier plus proficiency modifier for ranged), the damage the weapon does (including strength or dexterity modifier), and the damage type (i.e. slashing/piercing/bludgeoning).
As a barbarian, you have no spellcasting to write down.