In 2e, traps are there specifically to slow down the party and force them to be cautious. Even in 5e, traps are a break in the action and can be triggered before anyone sees them. The 2e style doesn't sound like your playstyle at all, but even the way 5e uses traps by default seems to not be your style either. So...

Tell them the trap is there

Traps are interesting when players interact with them. Tell them the trap is there! Yes, 5e has Passive Perception for this purpose, but don't let that ruin your fun if giving the check away "for free" will significantly improve your and your players' enjoyment of the game. (Passive Perception is still useful for surprise encounters and ambushes, after all.)

By telling them that a trap is present without telling them exactly where and what, you get straight to the interesting interaction, as they investigate the trap to figure out where it is, what it does, and how to circumvent or disable it. They may play it smart or they may make mistakes and suffer the trap's consequences. Some traps will be boring (and quickly dispatched) and some will be super-interesting (and be super-interesting). Either way—smart or disastrous, trivial or super-interesting—you are keeping the part of traps you find fun (the players messing around with them), eliminating the part you find frustrating (caution bringing the dungeoncrawl to a literal crawl), and quickly moving along to wherever the fun is actually to be found.

So skip straight to the fun, and tell them that there is a trap here. Let them figure out what kind and where exactly, but tell them:

As you move to cross the threshold, your keen adventurer's survival instinct tells you something is wrong. There's something not quite right about either the door's stone frame, or the room's floor… or maybe it's something else nearby that's off. Anyway, what do you do?

...and cue the flurry of investigation!

DCs for the traps, should then be converted like most DCs in 5e. 10 for easy, 15 for moderate, 20 for hard, 25 for very hard.

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.