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.
Using a Google search for "The Rules of Hidden Club," I found the document you're looking for as the first result.
It's essentially an explanation of the rules for the Stealth skill (found in the Rules Compendium or on the Compendium website) with a guide to applying them wisely.
The basics, heavily summarized, are:
The First Rule Of Hidden Club: Stay out of sight. If, at ANY TIME, you lack at least Cover or Concealment from an enemy, you lose Hidden against that enemy.
The Second Rule Of Hidden Club: Keep quiet.
The Third Rule Of Hidden Club: Keep Still. If you don't move more than 2 squares at a time, you have no chance of losing Hidden.
The Fourth Rule Of Hidden Club: Don't Attack. If you attack, you are immediately Expelled From Hidden Club, period.
The Fifth Rule Of Hidden Club: Don't let the enemy FIND you.
The Unspeakable Final Rule Of Hidden Club: When you do something to lose Hidden, you keep the benefits of being Hidden until the end of that action.
Best Answer
There is no Hidden Club in 5e
Hidden Club was invented to explain the tightly-interlocking mechanics that determined whether a D&D 4e character had the Hidden status. The very first premise of 4e's Rules of Hidden Club is the all-caps and repeated statement that everyone always knows where everyone else is. It's repeated and text-shouted because it's completely non-intuitive when you're coming to 4e fresh.
That's not the case with 5e. Coming from 4e it may seem self-evident that there must be a Hidden Club equivalent in 5e, but there isn't. That doesn't meant that there aren't rules, just that nothing like 4e's Hidden Club is necessary to explicate hiding in 5e.
How hiding works in 5e
The rules for hiding are deceptively simple. Deceptive, because if you expect them to be more complex, it's easy to unconsciously add in the expected complexity. But they really are quite simple:
These don't even really need numbered list items, because they're exactly how hiding works intuitively outside of RPGs. It's a convenient breakdown for the following rules cite sections though. Once we get through the citations that support those four points, we can dispose of them.
Interestingly, none of these depend on whether you're in combat or not, as we'll see.
1. If someone can see you, you can't attempt to hide from them.
This is the simplest cite. D&D Basic Rules v0.1, page 60 (sidebar):
2. Someone doesn't see you when they're not looking at you.
Same sidebar:
A creature needs to "stay alert for signs of danger all around" in order for it to only usually see you. The contrapositive's implication is that a creature who is not alert for danger all around does not necessarily see you, and whether they do or not is a matter of DM judgement of the situation. The rules-as-written for how creatures can visually detect enemies is therefore exactly the same as how real-life vision works. (Adding in fantasy forms of vision is a trivial exercise that doesn't alter how this rules text reads.)
Consequently, although this bit of rules text calls out combat in particular, the guidance it provides for how hiding works is not limited to combat. Combat is just a good example of a situation in which "most" creatures are "usually" on high alert and hard to escape the notice of.
Further evidence, from the same sidebar again:
A distracted creature as described is one not alert to danger coming from your particular approach, even if they're otherwise alert to danger. This dovetails with the intuition that the rules simply describe how sight, focused attention, and inattention work in reality.
3. Once hidden, you can be detected via sight, hearing, or both, so don't be seen or heard.
If you deliberately make noise or move from your hidden position, you are revealed (to those who are paying attention, as above). This is automatic (q.v.):
and
(The "usually" in there refers to the stipulations above, as previously established.)
But you can also be discovered simply by not being as quiet or as cleverly-concealed as you think you are (q.v.):
So if you do a so-so job of hiding your position or you're not staying absolutely silent, an opponent with keener senses than your ability at stealth will know you're there.
4. You can hide in plain sight, given a moment of being unseen or unattended and an appropriate concealment strategy
If you hide yourself before anyone knows you're there, your hiding spot can be in plain sight if your Stealth roll is just that good. Similarly, you can use a moment of distraction to conceal yourself, so that the opponent doesn't know where you hid when the distraction lets them attend to your general location again.
This is a pretty cool implication of the relatively simple rules. It's also still entirely in-line with the idea that hiding works just how it intuitively would in reality.
For example,
This kind of thing nicely matches how Arnold's mud-covering trick works in the first Predator movie, as well as Peeta Mellark's riverbed hiding spot in The Hunger Games. And it's mechanically simple to pull off in 5e with core rules, with no need for Feats like Hide in Plain Sight to be allowed to attempt it. You just need to set up a situation where you can try to hide in plain sight, and beat the Passive Perception of your anticipated opponents.
Of course being hidden by being in total cover or darkness is even better, but it's not the only way to be hidden, and that's pretty awesome for stealthy characters.
In conclusion, How Not to be Seen
So hiding is fairly intuitive: create or find a situation in which you'll either be overlooked by observers (harder to pull off, but common), or blocked entirely from their sight (easier to pull off, but needs suitable objects), and don't make any noise that they might hear. Then, hope that you're more skilled at stealth than they are skilled at noticing hidden things, and that further they won't come looking for you and actively test their perceptions against your skill.
Appendix: But hiding isn't all there is to Stealth
It must be emphasised that Dexterity (Stealth) covers more than just hiding — hiding is just one stealthy application. You can approach someone from behind quietly (q.v.), you can creep through shadows in a location completely visible to someone so long as they're not given reason to suspect a danger in your location and and you don't draw their attention (again: move quietly, move carefully). You can use Stealth to slip away from a dinner party without being noticed. You can use it to creep along a ceiling beam above some alert guards without making them look up. Stealth is much more than hiding.