As long as stats you need to attack in are 16's you're probably OK.
As you illustrate in your question you can get four stats to 16, though that will preclude any addition of feats (which are quite important for some classes, like the fighter, but they get more abil score increases so that's ok).
So no, MAD classes aren't exactly a problem in 5e. They still might be difficult to play, but the worst you're going to be is 10% behind everyone else.
The big factor here isn't actually the ability score increases, it's the ability score cap that is (or solves?) the problem. Basically, in previous editions (at least 4th), you started with a primary ability score of 18-20, and took it up to 30. And you only got to raise two scores twice every level up. That meant if you needed to raise 3-4 scores, you were going to have to choose between raising your top ability score less OR your secondary one even less (spliting increases between your secondary and tertiary) OR having your tertiary score fall way behind.
in BD&D, there is only really one (maybe two) PC classes that even use 2 stats, obviously the martial/melee classes in the PHB do as well. And with everyone needing CON, that really to me makes these characters MAD at least in proportion to other 5e classes.
At this point, though, MAD isn't a bad thing. As you illustrate, with a standard starting array, and the right race, you can have attack quality stats in 4 scores with only 5 ability increases. That's not to say you want to do that. I think most folks would prefer to get at least their primary up to 20, and that means forfeiting at least one of those 16s and no chance at feats. 10% doesn't mean much on any single attack, but the cumulative effect on your DPR will show. You might not feel it but it will be there.
Feats are an interesting quandry with MAD as well. Basically, supporting a MAD build will preclude the use of feats unless you are playing a Fighter or Rogue. So you must weigh the opportunity cost of the feats you won't be taking because you are choosing to use more than 2 abilities.
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).
Best Answer
This is really strong, at least for certain builds. Anyone who wanted to focus on Str anyway is likely to end up playing this race.
Firstly, Barbarians. Barbarians are currently one of the most MAD classes in 5e, needing good Str, good Con, and at least reasonable Dex. Your race would allow them to dump Dex and just focus on Str and Con.
Secondly, Druids. This is probably the worst case - your proposed race gives a Moon Druid an incredible boost to their Wild Shape forms. Str and Con are the strong points, and AC is the weak point, of most beasts, especially the ones Druids like to Wild Shape into.
Next, Fighters. This is the least problematic, since a Fighter could already afford to focus solely on Str and Con, but it does mean they end up with a better AC than they otherwise would have.
Oddly enough, it also enables Str-based Monk and Rogue builds. This isn't really a problem though, since in many ways Dex is inherently stronger than Str anyway.
Comparing it to Monk isn't really that useful, since they're essentially the same, but anyway:
Level 1: Typical Monk has 16 Dex and 16 Wis, giving them an AC of 16. A Barbarian of your proposed race has 17 Con and 15 Str, giving them an AC of 15, or 17 with a shield.
Level 4: Typical Monk has 18 Dex and 16 Wis, giving them an AC of 17. A Barbarian of your proposed race has 18 Con and 16 Str, giving them an AC of 17, or 19 with a shield.
Level 8: Typical Monk has 20 Dex and 16 Wis, giving them an AC of 18. A Barbarian of your proposed race has 18 Con and 18 Str, giving them an AC of 18, or 20 with a shield.
Level 12: Typical Monk has 20 Dex and 18 Wis, giving them an AC of 19. A Barbarian of your proposed race has 18 Con and 20 Str, giving them an AC of 19, or 21 with a shield.
Level 16: Typical Monk has 20 Dex and 20 Wis, giving them an AC of 20. A Barbarian of your proposed race has 20 Con and 20 Str, giving them an AC of 20, or 22 with a shield.
Your proposed race can always have a better AC than a Monk of the same level if they choose, despite the inherent weakness of starting with +2 in a single stat rather than +1 in two stats.
You're right that basing it off Dex and Con would be even stronger, but only because Dex is inherently stronger than Str. Your footnote about bonuses, however, is less accurate. A minor bonus to AC, while good for everyone, is hardly a game-changer. Your proposed feature, on the other hand, can form the foundation of a build. It replaces a core feature of the Barbarian or Monk, and is effectively completely free, especially since your proposed race has fire resistance on top of it.