[RPG] Do Characters lose skill ranks if their Intelligence score is drained below their base score

Short answer

The rules are not completely clear on this one, but in the absence of an exception, I would say that your Alchemist does get additional formulae if his intelligence increases permanently, and the first James Jacobs' quote you cite would seem to support this. This is no less 'illogical' than the fact alchemists get new formulae when they go up a level.

Long answer

The closest we get to RAW on this is under Ability Score Bonuses / Permanent Bonuses:

Ability bonuses with a duration greater than 1 day a­ctually increase the relevant ability score after 24 hours. Modify all skills and statistics related to that ability. This might cause you to gain skill points, hit points, and other bonuses. These bonuses should be noted separately in case they are removed. (PCR 555, my emphasis)

This is supported by James Jacobs' comment in the first post you mention:

All bonuses are retroactive when an ability score increases, be they bonuses to damage, to skill ranks, to hit points, to saves, to skill checks... all of them. Skill ranks not being retroactive are a 3.5 convention we specifically removed from the game because it was a weird exception to the rule, and since now there are no exceptions to this rule, there's no need to specifically state that skill ranks are retroactively granted if your Intelligence goes up.

Unless he forgot about spells/alchemists' formulae etc, no exception means no exception.

If you are looking for rules-based logic, I would say it is no more illogical for your Alchemist to gain a formula because his intelligence has permanently increased than it is when he goes up a level:

An alchemist begins play with two 1st level formulae of his choice, plus a number of additional formulae equal to his Intelligence modifier. At each new alchemist level, he gains one new formula of any level that he can create. (http://www.d20pfsrd.com/classes/base-classes/alchemist)

In terms of real-world logic you could just say that your alchemist had struggled with understanding the new formula, and it wasn't until his intelligence increased that 'the penny dropped'.

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.