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.
The Rules
Let's head back to the rules for what Perception actually does:
Perception. Your Wisdom (Perception) check lets
you spot, hear, or otherwise detect the presence of
something. It measures your general awareness of
your surroundings and the keenness of your senses.
For example, you might try to hear a conversation
through a closed door, eavesdrop under an open window,
or hear monsters moving stealthily in the forest. Or you
might try to spot things that are obscured or easy to
miss, whether they are orcs lying in ambush on a road,
thugs hiding in the shadows of an alley, or candlelight
under a closed secret door. (PHB p.178)
Finding a Hidden Object
When your character searches for a hidden object such as
a secret door or a trap, the DM typically asks you to make a
Wisdom (Perception) check. Such a check can be used to find
hidden details or other information and clues that you might
otherwise overlook.
In most cases, you need to describe where you are looking
in order for the DM to determine your chance of success. For
example, a key is hidden beneath a set of folded clothes in
the top drawer of a bureau. If you tell the DM that you pace
around the room, looking at the walls and furniture for clues,
you have no chance of finding the key, regardless of your
Wisdom (Perception) check result. You would have to specify
that you were opening the drawers or searching the bureau in
order to have any chance of success. (PHB p.178)
Passive perception is only mentioned in the context of:
- finding a hidden creature (PHB p.177 and, in the context of Surprise p.189)
- ether cyclones (DMG p.49 - how often will that come up?)
- detection secret and concealed doors (DMG pp.103-104)
- detecting pickpockets (DMG p.116)
- detecting traps (DMG p.121)
- chases (DMG p.253 - this is a simple extension/clarification of the hiding rules)
You would be quite within bounds to rule that these are the only things it works for; from your list, it would not help them find "secrets" (except secret doors) among other things.
Perception is for things that are hard to see
You don't need a skill check to see the sun or the floor or the clock tower; you just see them. Similarly, you don't need a perception check to see the orc unless he is hiding, however, see how not to be seen. Or to see the statue but maybe you do to notice the eyes are emeralds.
Things that can't be perceived
Notwithstanding, passive Wisdom (Perception) is of no use whatsoever if the object is totally obscured: treasure in a refuse pile, a trinket in a desk, a wand in a pocket etc. you have to actually go through the refuse pile, open the desk, feel in the pocket etc.; these all require active engagement; either active Wisdom (Perception) or Intelligence (Investigation) as you deem appropriate.
In addition, things that are lightly obscured give disadvantage on both passive and active checks; this includes darkvision or dim lighting. If something is outside the range of your bright light source then you are going to have to go in for a closer look.
Passive trumps Active
If you are allowing passive Wisdom (Perception) to be used universally, your comment that "... on average they get much better results with their passive perception" is irrelevant. If their passive perception is high enough to see the "thing" then they have seen it and active perception does not enter into the matter. If it isn't, then active perception will only help if they roll better than their passive perception.
Advantage
Helping someone look gives advantage on their active Wisdom (Perception) check; you can't do this with passive perception. Teaming the best Wisdom (Perception) people in the party with the worst more than doubles their effective perception.
Surprise
High passive Wisdom (Perception) only helps the individual character in a surprise situation. See How does surprise work in D&D 5e?; the relevant part being:
Surprise is something that happens to creatures; it does not happen to "sides".
Surprise is a quality of a creature; it does not describe a relationship between two or more creatures.
Essentially, your high passive Wisdom (Perception) PCs will rarely be surprised; everyone else in the party will still be surprised
Finding isn't knowing
DM: "You see a secret door; there are markings on the floor indicating that part of the left wall pivots into the corridor on the right hand side."
Player: "Great! I open it!"
DM: "How?"
Need I say more?
Player skill beats character skill
Let's say you have a valuable locket that you decide is hidden in a book in the third drawer of the desk. This is totally obscured unless the character opens the drawer; it cannot be perceived. Lets say you assign a DC20 Wisdom (Perception) check to find it if the drawer is opened. Lets also assume there is a trap door under the rug; this is not totally obscured because there are signs that the rug has been moved; say DC25.
Player: "I look around the room"
DM: "There is a chair, a rug, a desk with 4 drawers and a statue." If the player has a passive Wisdom (Perception) of 25+ you would add "It looks like the rug has been lifted and repositioned several times."
Player: "I search the room"
DM: Active check - if they get 25+ tell them about the rug. Depending on how you play the game "search the room" may encompass "open the drawers" - it doesn't have to but if it doesn't your players need to be aware of how specific they have to be. If they open the drawers then check against their passive Wisdom (Perception) to find the locket. If they search the drawers then they get an active Wisdom (Perception).
However, if they say "I lift the rug" then they find the trap door. If they say "I take out every item in the drawers and shake it" then they find the locket - no checks are necessary because these have just become obvious.
D*$king with the PCs
As you rightly observe in your 3rd paragraph, looking at the character sheets and penalising the players for the choices they made is unfair. D&D is a game of limited resources: maximising one skill means that something else suffers (lots of things, usually). The "world" is already getting this advantage; don't Nerf the things they are good at.
However, these traps and secrets were hidden by somebody, you can roll for how well they did with each one. Intelligence (Deception) is probably most appropriate, advantage can probably be assumed unless they were in a hurry. This gives you variation with the DCs without penalising the players. Note that this is exactly how Dexterity (Stealth) works.
Best Answer
Only when applicable!
First, let's keep in mind passive and active DCs can vary. From LMoP:
So, passive scores acting as a floor to a roll happens only when the DCs match (otherwise, no need to have different DCs). For the remainder of this answer, let's assume the DCs for active checks and passive scores match.
Now, when can we use passive scores? The rules are clear.
In the case of Perception, it is an ability that's always being done repeatedly by adventurers, and is described by the Hiding rules. As other answers have mentioned, about spotting hidden enemies, passive Perception is the floor:
But can be superseeded by a better roll with an active search:
That being said, I think it makes sense for some other scores to work passively. For example, Insight is my prime choice, as I assume PCs will notice by default if they're being lied to. When new monsters appear, Intelligence passive scores also tell PCs lore and names of enemies. If PCs then ask for active actions, then they may supersede the passive score.
However, actions like picking a lock, they're a short set of actions that perfectly match an active Sleight of Hand check. If he can just try and pick it all day, then I consider the passive score, but when they reach the house for the first time and try to break in, it's an active check, and not superseded by any passive scores.
Passive score first (if applicable). Then active check.
In the end, it depends on DM and PC. Use passive scores when it is something the PC is assumed to do by default, repeatedly, with no consequence (remain alert, ready to fight, recall information, maybe even seduce NPCs to affect their mood), and these may be superseded by active checks if PCs ask for them. Whenever it is not a repeatable task, and instead an actual event with consequences for failure (maybe as simple as wasted time), use active checks, since passive scores are not applicable.
Don't just assume all skills have a minimum. It would cause an 11th level Rogue skill to be useless.