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.
RAW the Rogue does receive the benefits of Expertise to his passive.
A passive check is a special kind of ability check that
doesn’t involve any die rolls. Such a check can represent
the average result for a task done repeatedly, such as
searching for secret doors over and over again, or can
be used when the DM wants to secretly determine
whether the characters succeed at something without
rolling dice, such as noticing a hidden monster.
Here’s how to determine a character’s total for a
passive check: 10 + all modifiers that normally apply to the check. - Player's Handbook P.174
The player has invested in making this character good at this, this is not a bad thing.
Rogues in general are supposed to be good at this type of thing, hence the proficiency bonus class feature and the player specifically spent a feat on making his passive perception better. The player could've spent this on another feat to specialize in another area (or gain a combat ability). This is a good thing, while it may make it difficult for the Rogue to be surprised by a trap or an ambush you should in no way try to outmaneuver this.
Don't worry, there are downsides to the Rogue for this.
To reliably spot traps and ambushes the rogue will need to be at the head of the marching order so that he has clear sight lines. Rogue's are not particularly hardy and the party as a whole is inviting more risk for this reward. Likewise while he is extremely excellent at spotting physical dangers, the Rogue will not be able to detect magic wards and other dangerous enchantments and may equally blunder into them.
Ways to handle his detection of a trap or an ambush
You are correct in assuming that only that PC has seen the trap/enemy. Unless he has some mode of telepathic communication he will need to speak out and announce the threat to everyone. The best way to handle this sort of thing is to pass notes or send text messages to the player(s) able to see/detect the issue and leave it to them to react and tell someone else. Intelligent enemies will see/hear the rogue warning the party and the combat should start immediately (no surprise round though).
The PC does not become omniscient of the trap upon detecting it.
The PHB itself is very, very vague on what information is received when a PC detects a trap. However, the DMG does have a nice little section about traps, their detection, and disarming them:
If the adventurers detect a trap before
triggering it, they might be able to disarm it, either
permanently or long enough to move past it. You might
call for an Intelligence (Investigation) check for a
character to deduce what needs to be done, followed
by a Dexterity check using thieves' tools to perform the
necessary sabotage...
...In most cases, a trap's description is clear enough
that you can adjudicate whether a character's actions
locate or foil the trap. As with many situations, you
shouldn't allow die rolling to override clever play and
good planning...
...Foiling traps can be a little more complicated.
Consider a trapped treasure chest. If the chest is opened
without first pulling on the two handles set in its sides, a
mechanism inside fires a hail of poison needles toward
anyone in front of it. After inspecting the chest and
making a few checks, the characters are still unsure
if it's trapped. Rather than simply open the chest, they
prop a shield in front of it and push the chest open at
a distance with an iron rod. In this case, the trap still
triggers, but the hail of needles fires harmlessly into
the shield. - Dungeon Master's Guide p. 121
Essentially though whether trap disarming is simply a dex check, a series of checks, and/or involves serious RP is up to you as a DM. I would encourage the open-ended approach the book suggests as it adds complexity and makes trap checking and disarming a more engaging process. Sitdown with the party OOC and discuss what the table as a whole thinks should happen for trap checks and move forward based on that consensus.
Best Answer
Yes, Expertise affects passive Perception checks.
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: