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.
Yes, it's a pretty obvious secret door.
As you say, passive Wisdom (Perception) is 10 + WIS, so anyone with a non-negative WIS modifier will passively notice it.
But that doesn't mean you definitely see it.
The interesting exception I see is if your party is rushing through the area. Per Travel Pace moving 400'/minute or 4mi/hour will earn you a -5 to passive perception (PHB p.182). Lots of characters don't have the +5 WIS modifier necessary to offset that, and might blow right past it.
@Dale M points out the more likely scenario when this may happen: disadvantaged passive checks. By rule (PHB p.175), a passive score receives a -5 bonus when subject to disadvantage.1 Low light, obscuring conditions, or other environmental challenges could create disadvantage, similarly challenging one's ability to notice a DC10 secret door.
1 The "disadvantage=-5" rule of thumb is a lamentable addition to 5e culture. But, as this excellent answer shows, in the specific case of DC10 it's the perfect modifier. The second graph in the linked answer -- the vertical distance between the black and blue curves, to be precise -- shows this to be the case.
Best Answer
Passive Investigation could be a way to determine whether to feed players information that their character might pick up on, but the player might not think to explicitly ask. Note that a passive check can be against a DC, rather than an opposed roll. Some examples:
How are these things not Perception? Because what you are noticing isn't remarkable unless you compare it to your expectation of how things should be. It's a test of reasoning rather than of, well, perception. I keep wanting to say "think Sherlock Holmes", but he's also rocking Perception, so that's actually a terrible example. So... Rain Man, maybe?
PHB suggests passive checks (p 175) where the GM doesn't want the players to know they've succeeded or (more typically) failed at something.
Hidden checks are fun:
A paranoid player might call for a check for all of these things, but if they don't think to, they won't know they failed until you start giggling.