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.

How do the Kobolds remember which parts are trapped?

Basically, this answer is about weaving the Kobold's own marking system into the narrative. It does assume you draw your own maps and don't use Dungeon Tiles or anything.

Obtain 6 or so pretty looking symbols (they don't need to have meaning, but if they look Draconic it's bonus awesome) Mark every square of the map with one of them. In the narrative, explain that the Kobolds did just that; they covered all the floors with all sorts of markings. For each room, assign 2 symbols to mean "trap" and the others to do nothing. Vary the symbols per room.

This will clearly signal to the players "this room is trapped", but then the word Kobold is basically a synonym for "traps" anyway, so that's okay. It will keep your players attention strongly on the traps, they will try to figure it out by watching Kobold movement (which is good! traps exist in the narrative to be interacted with) and will mostly stumble right into a trap in room 2 before they realise that the Kobolds vary which symbols mean trap in every room.

You could even tie a meaning to each symbol (in Draconic) and explain the words to players who speak it, and then have the chosen symbols make sense per room. Since Kobolds employ deadly traps in their home, they need a system to keep down their own losses, and that has to be simply enough to teach to their kids while still being confusing enough that confound and/or kill enemies.

To tie it even stronger into the narrative, include a room that the Kobolds use for training their young (probably one with few or no Kobolds in it, since they know it's less hurtful) where the traps just drop a bucket of water on their heads or launch sticky paper at them.

And maybe in the newest room, the Kobolds haven't had time to place traps yet, but they did place the markings. Your players will be properly trained by this point to move very carefully, figure out the system, and probably be amused when they realise the Kobolds have trained them like Pavlov's dogs to avoid certain squares, even if they don't do anything. (If they are trained so well they never trigger any, make sure they find a scrap of paper later that has instructions on which traps should go where that makes it clear none of them have been installed yet. Remember: if you don't tell them, it doesn't exist.)

(Regarding the last part, I once trapped my players in a room inside the Temple of the God of Theater, by giving them a huge grid full of letters whose only effect was that certain letters made the floating balls of electricity in the corners light up ominously. They were told that the room was "very dangerous and none had ever traversed it". It took them an hour to figure out they were being played and they loved it.)