Pathfinder is pretty straight forward when it comes to boni and penalties, and we hage a Word of God:

A penalty DOES effectively lower an ability score, but it's temporary. When we have an effect that does damage or drain, we're intending that effect to last as long as it takes to heal the damage by outside sources. When an ability score reducing effect has a built in duration, after which the reduction goes away, we use "ability score penalties" instead of damage.

James Jacobs, Creative Director [at Paizo]

Therefore:

1. Take the Ability Score as basic number
2. Apply all relevant permanent modifieres, like enchantment bonus and drain, this is the Modified Ability Score, it will determine our 'Base' from where to modify further.
   1. If you are at an ability of 0 or negative: Roll a new character...
3. Sum up all temporary: boni, then damage and finally penalties (as negative numbers), one after another, apply the following on the way:
   1. Should drain + damage get you down to a score of 0, nasty things happen (unconscious, crippled, etc)
   2. Should a penalty try to lower the number to 0 or below, set it to 1.
4. Unless you hit a branching case (which is either dead, icapicitated or a score of 1 with bonus of -5), calculate the current Ability Score Bonus from the Modified Ability Score:
   • round down the result of \$\frac{\text{Modified Ability Score} - 10}{2} + \frac{\text{Sum of Boni, Damage and Penalties}}{2}\$
   • However RAI from the FAQ seems to be more the recalculation of the Ability Modifier each time with each Score: \$\frac{\text{Modified Ability Score}+\text{Sum of Boni, Damage and Penalties} - 10}{2}\$

So, you have strength 10, no strength damage and drain. Now you apply no bonus and a penalty of 5. The resulting value is Strength 5. Strength 5 results in an Ability modifier of -3. Would you have Strength 11, the modifired strength would be 6, and the Ability Score Bonus thus -2.

Now, your other question is similar:

You have 5 Ability damage and a 5 Ability Penalty applying to the same Ability. Here it gets a bit tricky. First, we check the Base score again. 10. Apply the bonus (0), then drain (0), then damage (5) and finally a penalty (5). After the damage, the Ability Score is 5. Applying an ability penalty of 5 to an ability of 5 is not possible, as Ability Penalties may never drop a score to 0, but instead they drop it to 1 instead.

This is played straight as just tracking them and then stacking them, as seems to be the general voice on the paizo forums:

You track each [penalty] separately and they stack. ... They don't actually reduce your score. Ability drain on the other hand, however DOES actually reduce your score.

... that ray of enfeeblement couldn't have reduced your goblin's strength to 0 (seen here)

also

Penalty and Damage don't lower the score, only Drain. Because of that, your score stays the same for the purpose of required stats.

The reference to lowering ability scores below 1 only applies to temporary effects, such as those from Ray of Enfeeblement, but not from diseases (which last permanently or until removed). You can't be rendered unconscious or dead by a temporary penalty from a spell, but you can by a permanent penalty from a disease or curse. (seen here)

also

ability modifiers are a summed modifier, not a separate set of individual bonuses or penalties. Ability damage doesn't directly reduce the relevant stat but it follows a similar logic. By this I mean: gaining 4 ability damage is a -2 penalty, not two -1 penalties. These are mathematically distinct concepts in pathfinder rules. (seen here)

but:

[That the same spell can't stack with itself] comes from page 208 of the core rulebook:

"Spells that provide bonuses or penalties...usually do not stack with themselves." (seen here)

Now, your test cases...

• Score 10, 5 Penalty, 5 Damage

The sum of the reductions is 10, but some of it is a penalty. As Penalties can't reduce to less than 1, the score is 1.

• Score 10, 4 Bonus, 11 Damage

As long as the Bonus of 4 stays, your score is 3. As soon as it fades, you go to -1 with all consequences, till the damage is healed to some degree.

• Score 10, 4 Bonus, 11 Drain

Your Modified Ability Score is -1. You may never have an Ability Score that is -1 from permanent sources. Drain is permanent. You die instantly.

• Score 10, 4 Bonus, 5 Penalty, 9 Damage

Sums up to an effective 10 reduction of the score, BUT, there is still some Penalty (1 point!) of it. Penalties still can't reduce to less than 1, so score is 1.

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.