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.
Seekers
People on watch for eight hours or so are not hyper-vigilant all the time - they chat, warm their hands at the fire, go for a leak etc. Unless there is some reason that seekers are actively looking for the hiders (e.g. a recent alarm) then this should be a passive Wisdom (Perception) check so in effect the hiders are rolling against a fixed (but unknown) DC.
If there are multiple seekers then they should us the Working Together rules on p. 175 of the PHB; this gives advantage to the person with the best Wisdom (Perception), +5 on passive or 2 rolls on active checks. Don't forget that there are lots of ways that this advantage can be cancelled: dim light being the most often overlooked.
Hiders
The needs of your scenario should dictate if you use group checks or not. That is, is it more interesting/fun for the characters to succeed or fail as a group or individuals.
Group checks (PHB p.175) make things much easier for the hiders. For example, for 4 characters all needing to roll an 11 or more a group check will result in success 68.75% of the time whereas individual checks will have no one detected only 6.25% of the time (http://anydice.com/program/8892). Notwithstanding, bigger groups make both more difficult, however, for group checks it is a slow decline while individual checks fall off a cliff.
Helping someone sneak is problematic, I can see arguments both ways. Obviously, calling out instructions would be counter productive but relying on others to watch the guards while you only watch for their hand signals would really help. My personal feeling is if the players can tell you how teamwork gives them advantage then let them have that advantage. (I am assuming that it is usually the players that are hiding, not the monsters).
Best Answer
It's possible, but some important differences between Stealth and Perception make it questionable
The rules on passive checks give us some guidance here. As you mentioned (bold added):
Although this isn't necessarily an exhaustive list of when Passive Perception would be used, it's the best guidance we have. So let's go through it point by point.
Average Implies Many Tries: What About Failures?
The checks that explicitly have passive versions (Perception, Investigation) are ones which you are essentially taking myriad times every second. You are always noticing hundreds if not thousands of things, most of which your mind edits out. And you are always thinking, even if you try not to. In this sense, they are similar to stealth, since stealth is more of an ongoing process than a single moment's effort.
However, where Perception and Investigation differ from Stealth is that many of these myriad checks can fail without causing the overall effort to fail, as long as some of the checks succeed. If you are walking towards an ambush, and for two seconds don't notice it, you've still "succeeded" on perceiving it if you then notice the ambush on second number 3 (and can react to it before the trap is sprung). However, if you stealthily creep forward for two seconds, and knock over a shelf of fine china in second 3, then stealthily creep forward on second #4, I think it's safe to say that your subsequent stealth won't make the effort a "success."
To summarize, when a passive check is used to represent an average result, it should be done when a failure on the check would not make a subsequent effort harder. If you fail to pick a lock, you may break or twist its mechanism. If you fail to convince someone of something, they may get tired of your attempts to sway them. But if you fail to see something, that doesn't make it less visible later. And if you fail to be silent, you can't make someone un-hear you.
What about a hidden observer?
The second point is a more viable one. A passive check could be used when a DM does not want to reveal to players that there is something happening which could succeed or fail. It's totally valid to suggest that there are times when a character might be observed secretly, and the DM doesn't want to let them know. Perhaps a seemingly inert gargoyle above is really a careful sentry, or an invisible guard is on watch. For whatever reason, it's quite possible that someone could be listening or looking for a character without them knowing.
But most of the time that a character is being stealthy, they only think they might be observed. If you only attempted to be stealthy once you'd actually seen a guard, then the guard would usually have seen or heard you already as well. Rather, most times that people attempt to be stealthy, they know that there might be an enemy ahead, and respond accordingly.
So although a DM certainly has the prerogative to assign a "passive stealth" to a character (so they can hide the fact that an observer is in the area), they risk blurring the line between situations that merit a passive check and ones that do not. And in doing so, they risk setting a "floor" to the standard stealth check, by basically allowing a passive check most times an active one would also be valid. We'll go into this in our next section.
How stealthy must you be to be stealthy by default?
Running with the "unseen observer" angle for a moment, let's assume that a character (who for the sake of simplicity, we'll assume is a Rogue) is not in a situation where they think they might be observed: not in a dim dungeon or mysterious cave, but rather walking down the street of a familiar village, or climbing the stairs of their own home. In these situations, a Rogue would be unlikely to be making active stealth checks. But a passive check could be useful to a DM, to hide the lurking menace and maintain the surprise.
Now, this argument assumes that a stealthy character is stealthy by default, which is not that much of a stretch. After all, it's easy to picture how a trained Rogue might be light on their feet during everyday activities: making only a muffled footsteps as they traversed their own home, or seeming to appear suddenly to a group of friends. These activities might be second nature to a Rogue, the same way that a brawny Fighter might always lift heavy loads with athleticism and grace, even if they weren't trying to show off.
However, it's worth asking exactly how trained or experienced a Rogue needs be to attain this particular level of mastery: where even without trying, they are notably competent at their common skills. And there is an in-game answer to this question: 11th level.
An 11th level Rogue gets this powerful advantage. To assign a Passive Stealth result before 11th level, then, might be over-valuing the skill level of the character. At the very least, it would be stepping on the toes of an 11th level ability, many of which provide class-defining advantages (especially since there's no way that one character can have 11+ levels in more than one class). As such, giving someone an ability that is similar to an 11th level class ability (which they have not earned) is something I'd highly advise against, without serious thought.
You might be missing out
If you are the DM of a game, and you think that a passive check is warranted, then you are correct: it is the DM's prerogative to call for checks, active or passive, at times they deem appropriate.
But it's worth asking yourself what the passive stealth check is for. If it's for determining the average result of many stealth checks, then consider that those "many" checks may have some failures in them. And if it's for hiding the need for a stealth check from players, keep in mind that stealth, by its nature, is usually done when its need is uncertain.
And perhaps just as importantly, ask yourself what you gain and lose by a passive stealth check. Consider that a "passive perception" check will prevent the players from feeling like they are missing out: after all, if you call for a perception check, and then declare they see nothing, a player will likely feel cheated (perhaps there is something here, but I'll never know). But if a player is asked to make a stealth check, and then told nothing happens, how are they likely to feel? I suspect that the check will either add to a feeling of competency (if the player believes they succeeded) or a feeling of suspense (if they believe they failed). Either of those feelings is a valuable addition to many games: so ask yourself if it's worth losing out on them before you decide on a passive check.