Proficiency in a skill is used when... you are proficient in that skill. The exact amount changes by level. See the chart in the game rules, but it starts at +2 and eventually goes to +6.
From your example, the fighter (with a wisdom of 12 or 13, for a +1 wisdom modifier) has a perception modifier of +3, so that example character must be trained in perception. Perception is on the fighter's list for potential proficiencies (choose two from a list), and clearly the example character did choose that (or got it from a background choice).
Passive perception is not 14, because the passive score is just 10 + your regular bonus for that skill, not 10 + your ability modifier + your regular skill bonus which already includes your ability modifier.
The dwarven cleric isn't trained in perception, but has a high wisdom, so the +3 wisdom modfier = the skill modifier too (no proficiency bonus is added) and the resulting passive score is 10 + just the ability modifer — so, also 13, but for different reasons.
In general, the proficiency modifier is used when you are trained in (or, in other words are proficient with) the skill, tool, or weapon in question.
This is explained in Chapter 7 of the freely-downloadable basic rules, and the sidebar about Hiding on page 60 explains all of this pretty nicely. In the official terminology, there does not appear to be a name for the number you write next to the skill, other than the relevant ability modifier plus bonuses and penalties. But, colloquially, I think many people will use the 3rd edition terminology and call this a "skill modifier" or "skill bonus", or specifically "perception bonus". People will say "perception skill" too, although I think that should be avoided because it could arguably mean either the bonus, the passive score, an actual roll + modifier, or even the just whether you're trained or not.
Yes, Expertise affects passive Perception checks.
Passive Checks (PHB 175)
Here's how to determine a character's total for a
passive check:
10 + all modifiers that normally apply to the check
[...]
For example, if a 1st-level character has a Wisdom of 15 and proficiency in Perception, he or she has a passive Wisdom (Perception) score of 14.
Simply put, if you have a WIS mod of +3 and a proficiency bonus of +4 (with proficiency in Perception), your passive Perception score will be 10 + 7, or 17. If you have Expertise in Perception, the +4 is doubled, so you're correct in judging that your passive Perception score will be 21.
Even better, if you're in a context that would grant you advantage on your Perception check, your passive Perception would be no less than 26, thanks to how advantage modifies your passive Perception score:
If the character has advantage on the check, add 5. For disadvantage, subtract 5.
Best Answer
Aboleths do have [something which is mechanically equivalent to] Expertise in those skills
The Monster Manual uses the word 'expertise' when describing how certain creatures might have higher-than-expected bonuses to certain skills:
There's a lot of contention over whether this translates to such creatures literally having an unwritten feature, Expertise, in these skills (same as the Class Feature that Rogues and Bards have access to) or if this is just colloquial language (see here for more), but regardless, it's clear that monsters can and do have something which is mechanically equivalent to Expertise in certain skills.
As for why...
The Monster Manual entry for Aboleths has a lot to say on the subject, but one passage in particular stands out...
So it is clear that Aboleths having enhanced Intellect, and in particular, enhanced memory, is well justified in the lore of the game, and is thus appropriately reflected in the game's mechanics.