Okay, this is a lot. In order to actually be able to answer it without writing an essay, I won't make complete comparisons for each subrace. As I mentioned in a comment, if you want a detailed feedback, I would create a separate question for each subrace - they are too different from each other.

Your Base Race has almost no features, which is fine considering the main thing is on the subrace. It's still worth to note that consistently the base race is the one giving a +2 AS bonus.

Earth

You get a +3 AS, which is the usual for playable races. +2 Str +1 Cha is an awkward distribution though (same as dragonborn), probably most useful for Paladins (usually the class Dragonborns are played). For most of other classes, one of the two stats will be kinda wasted. By itself, this doesn't make the race weaker. Tiefling gets an even worse bonus distribution.

Barkskin is a 2nd level spell. Not only you are giving a free 2nd level spell to the character, you are giving a free 2nd level spell slot as well. We could enter the discussion if barkskin is a strong spell or not (my opinion: it's not, actually it's weak), but the fact is: 2nd level spells to 1st level players is alot.

I would change it to a feature that might scale with PC's level, similar to Breath Weapon from Dragonborn.

How strong will your resistance to piercing be depends on the campaign. While Dragonborn and Tieflings have stronger resistances (elemental), they are more useful in later stages of the game, while piercing might be too strong in early stages, where physical damage is most common.

I would say the Earth subrace is (maybe too) strong in Tier 1 adventures, but becomes balanced from Tier 2 onwards. I'm not sure about Tier 3, but it becomes kinda weak in Tier 4, compared to the playable races.. One easy way to fix it, as I mentioned, is making the Barkskin feature scale with the tiers. I will make a note here that is valid to the following feedbacks, though: The bonus from races usually are less important in the endgame anyway, since your class features and magic items will compensate for it.

Air

Similar to Aarakocra. I am not sure the +10 move speed in land and Hover compensate both the -1 Dex and -20 Move speed flying, though. Also, I'm not sure how mechanically useful the weather prediction thing will be, that depends on you. Also, while Aarakocra +2 Dex +1 Wis helps alot with Rangers and Druids, the +1 Cha instead makes it a little more awkward, probably being more useful for Bards. Sorcerers might be interested in the flying feature, though, and get more from +1 Cha than +1 Wis.

As it is, it's clearly weaker than Aarakocra. (Yes, I know you asked "compared to PHB", but since PHB doesn't have flying races, I'm forced to compare it to EE, sorry). That said, it isn't completely underpowered since the AS bonus is more attractive for bards and sorcerers.

Fire

My first impression was god it is too strong. Then I noticed that, while you added a lot of powerful features together, they don't have too much synergy.

The main problem is that, from internal comparison, unless Flying can compensate it very well, it's way stronger than the Air version.

Other than that, it's very similar to a Red, Brass or Gold Dragonborn, changing the Str for Dex.

In my opinion, having the Fire Bolt is weaker than the Breath Weapon, even if it can be cast as many times as you want, because you probably will have other ways to deal damage anyway. Thinking in Dex based characters, they will probably be shooting arrows instead of fire bolts. The Breath Weapon is AoE, though, and usually cast by a class that doesn't have that much AoE early (Paladin), so it's more useful in the specific situation it's actually used.

The Grappling damage would also be more useful if it was a +2 Str bonus instead. Building a grappling character without a Str bonus is too suboptimal. I can't even guess how strong this feature will be as it is right now, sorry but you'll have to playtest.

So, comparing to Dragonborns, it will depend on how much Darkvision is a think in your setting. If darkvision is really important, it's balanced compared to Dragonborn.

If we compare it to Tieflings, it seems to get outscaled. Early on, you have a damage cantrip and a grappling feature, while Tiefling has a flavor cantrip. From level 3 onwards they get spells, though, while the scaling of this subrace is only based on the fire bolt cantrip. Again, even if it gets outscaled, it's not an obvious choice since your AS bonus is more attractive to Dex users.

Still, I would fix the fact that the AS bonus (dex) and the features you are giving don't synergize well.

One possible solution is to change the Fire subrace to +2 Str and the Earth to +2 Con instead (if you don't want both to have the same AS bonus). It would fit the "durable" flavor of the Earth/stone and the "aggressiveness" of the Fire.

Note that, if you do, the Fire subrace becomes a very strong option for grappling buildings. I wouldn't say this makes it overpowered, though, since grappling builds are usually more for fun than optimal.

Water

This one is hard to compare. The Healing feature seems weak (just Your Level is not much HP. Compare with Dwarves getting +1 HP/level, which is arguably stronger). Create/Destroy water is a 1st level spell, but this one doesn't need much discussion about being strong or not - it's not strong mechanically.

So it's about the other 3 features. First, the AS bonus: +1 Con +1 Cha is weak. Variant humans get to choose their two +1s and get a Feat on top of that. But they are arguably the best race in the game, so let's not do this. Still, +2 AS is underwhelming and I would expect the features to compensate it strongly. For the Air case, flying might do it, let's see the Water.

Water Breather again will obviously depend on your setting. It's arguably harder to use than Darkvision or Flying, though, since while your party can easily fight in a dark cave or an open field, fighting under water is way more unlikely and usually not desired by PCs.

That leaves Armor of Agathys to compensate the underwhelming AS bonus. Well, sadly it doesn't. It is a very, very strong feature in the first two levels, where +5 HP and 5-10 damage to the enemy are alot, but it quickly becomes weak as you progress in the game. Again, I feel you are lacking scaling. Increase these numbers as the tiers go up. So, similar to the Earth subrace, I'd say it is (maybe too) strong in Tier 1, but gets weaker (and even underpowered) in Tier 3+.

TL;DR

My main criticism here is that all your subraces lack scaling. While Air is fine without it, since Flying itself has a constant utility through the game, the other subraces might suffer from it. It is fine if the campaign is focused in Tier 2, but they seem too strong for Tier 1 and too weak for Tier 3+. Again, note that being too weak in Tier 3+ is not that important, since you have lots of class features and magic items to compensate for that. Still it's something you might want to take into account.

I suggest scaling as a way to fix it, but introducing mechanics that have utility through the game (such as having advantage on something, proficiency on something, etc) also works.

It's worth to mention, though, if you are playing a long campaign that goes through, for example, levels 1 to 15, your subraces are mostly fine, as they will be strong early on and weak later, and it's a valid trade-off for your players to choose. I have linked a chat discussion about this, if you are interested.

Side note:

If you are wondering why I'm not comparing to Dwarves, Elves, Gnomes or Halflings, which also don't have numbers scaling with the levels, it's because they have proficiencies in skills, weapons or tools, or advantages in saving throws, which are the things scaling. Also, they get lots of other features, such as Trance, Luck, etc. Comparing to Dragonborns, Tieflings and Aarakocras is easier and more fair (imo) because they are closer to what OP has made. It's hard to compare if proficiency in perception, using daggers and sleeping 2 hours less is better than 1d8 damage when grappling.

Combat-wise, it feels a bit much.

I recently started a druid. My first intent was to find a way to remain effective in combat despite depleted resources ; I found out that the druid hasn't much to offer, in that regard:

1. Ranged options (such as sling attacks, or produce flame) lack range and damage.
2. Melee options (such as shillelagh or thorn whip) lack damage in the long run.
3. AC is often on the low-end - unless your DM is generous on non-metal armor distribution, or you focus on dexterity.

It is indeed very frustrating, and encourages battlefield control over direct damage. Your homebrew subclass fixes all 3 issues, including strong blaster, healer and tank features: it would have been a no-brainer for me, and that's a warning for balance.

Bladesinger comparison

There are not that many full spellcasters with Extra Attack, and Bladesong looks very close to Heart of Fire. It may seem balanced... until you dig a little more:

\$\begin{array}{|c|c|c|} \hline \textbf{Ability} & \textbf{Bladesinger} & \textbf{Circle of flame}\\ \hline \text{max AC using mundane gear} & 23 & \textbf{25} \\ \text{Requirements:} & \textit{mage armor}\text{ spell} & \text{shield, class level 18} \\ & \text{high dexterity & intelligence} & \text{high dexterity or nonmetal half-plate} \\ \hline \text{max AC using magical items} & 25 & \textbf{31} \\ \text{Requirements:} & \textit{robe of the archmagi} & \text{+3 shield & armor, class level 18} \\ & \text{high Dexterity & Intelligence} & \text{high Dexterity or nonmetal half-plate} \\ \hline \text{Hit dice} & \text{d6} & \textbf{d8} \\ \hline \text{Damage control} & \text{reaction & spell slots-based} & \textbf{healing spells} \\ & \text{Damage reduction} & \textbf{reaction-based healing} \\ \hline \text{Concentration}& \text{+Intelligence modifier} & \text{better AC = fewer concentration checks} \\ \hline \end{array} \$

All in all, the Circle of the Flame druid vastly outperforms its cousin - and can even do so with little ASI investment, should you find a proper armor.

Circle of Land & Circle of Spores comparisons

The Circle of Spores, from Unearthed Arcana, is probably an attempt to offer wannabe-melee druids an alternate to the circle of the moon's combat forms. Let's include it here:

\$\begin{array}{|c|c|c|} \hline \textbf{Subclass features} & \textbf{Circle of the Flame} & \textbf{Circle of the Land} & \textbf{Circle of Spores} \\ \hline \text{level 2} & \textbf{$fire\ bolt$} & \text{one druid cantrip} & \textit{chill touch} \\ & \textbf{1-minute huge melee buff} & \text{Natural Recovery} & \text{10-min small melee buff} \\ & & & \text{reaction-based poison damage} \\ \hline \text{level 6} & \textbf{Extra Attack} & \text{ignoring difficult terrain} & \text{1-hp zombies} \\ \hline \text{level 10} & \textbf{+Wis mod to fire, concentration spells} & \text{various immunities} & \text{AoE spores} \\ & \textbf{fire resistance} & & \\ \hline \text{level 14} & \textbf{turn fire/lightning damage into healing} & \text{beasts & plants hesitate to attack} & \text{various immunities} \\ \hline \end{array} \$

It looks like the Circle of the Flame has the best features at each level: best cantrip to allow ranged combat, best melee-buff, extra attack is sick, enhancing concentration-based spells is unique... Even the last feature, that appears situational at first, becomes cheesy once you start to fuel it with your own AoE spells.

Suggestions

So, how do we fix this?

First, you can't aim for balance, and be a strong healer, blaster and tank at the same time. I removed the healing capabilities, as the tank part seemed your main concern, and the theme calls for improved fire damage.

Second, on the AC question: "no metal armor" and bounded accuracy make things difficult. If your DM rewards you with powerful nonmetal armors, he shouldn't allow big AC temporary bonuses in an homebrew class. The opposite is also true: if he allows big AC temporary bonuses... he should make sure you keep a lowly hide armor for your entire career. Hence the solution offered here - which seems the good way to reliably get good AC, without over-investing in dexterity or "forcing the hand" of your DM.

Here's the result:

Only You

When you choose the Circle of Flame at level 2, fire bolt and green-flame blade become druid cantrips for you. You learn an additional druid cantrip of your choice.

Leaves choices, and includes another fire-themed cantrip. Green-flame blade is mostly aimed at tier 2+ melee: extra attack was too much (especially if you intend to add Polearm Master), but this cantrip offers something comparable to the cleric's Divine Strike.

Heart of the Flame

At 2nd level, you gain the ability to awaken the primal spirits of flame. As a bonus action, you can expend a use of your Wild Shape feature to awaken your inner flame, rather than transforming into a beast form. While this feature is active, you gain the following benefits:

• You gain 3 temporary hit points per level you have in this class
• You shed bright light in a 30-foot radius and dim light for an additional 30 feet.
• You add your Wisdom modifier to concentration checks.

These benefits last for 10 minutes, or until you become unconscious.

Temporary HP is probably the way to go to add survivability to a druid: it matches what has been done for the Circle of Spores and Circle of the Moon. I removed the AC boost: if you insist on keeping it, you'll have to either keep it low (like +2), or remove the ability to use a shield - if you care about bounded accuracy, of course.

Purification through Fire

Starting at level 6, you gain resistance to fire damage. You also learn a purification ritual that allows you to prepare metal shields & armors, so that you can use them.

With your average 14-dex druid, it means up to +3 AC - and an easier access to magic armor in the long run. It is a game changer for most tables, where nonmetal armor is scarce - and can be balanced by making the ritual as simple or as complex as you wish (involving time, gold, crafting skills, a risk to destroy the item?).

Heat of Battle

Starting at level 10, once per turn, whenever one of your spells inflicts fire damage to a creature, you can add your Wisdom modifier to that damage.

Idea is to open that damage buff to more options (e.g fire bolt, green-flame blade) - while keeping it at a reasonable level (only one creature per turn).

Flaming Soul

Starting at level 14, you gain immunity to fire damage. You can also apply your heat of battle on a second damage roll, each turn.

Either affect both targets of green-flame blade, or use it on both flaming sphere and your cantrip of choice.

I'm absolutely not sure that this is well balanced:

• I feel Nature and/or Forge Domain clerics might feel cheated with this, as some of their core subclass features are included here, with few drawbacks
• having both a better AC and temporary HP, while keeping access to healing spells, means that you are tanking pretty hard
• Extra attack was too much - and this may also be. Green-flame blade + Heat of Battle + shillelagh makes a strong combination.
• Bonus to concentration might go out of control if you go for the War Caster / Resilient (Con) route. Replacing it with Con save proficiency or advantage on concentration checks may help avoid those shenanigans.

But I do think it is closer to "balanced", when compared with Land & Spores druids. Obviously, feel free to switch things around (access to metal armor as soon as level 2?) - and take or leave whatever fits best your idea.