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.
TL;DR
The rogue makes a single Dexterity (Stealth) check against a DC15 passive Wisdom (Perception). Beat that and the rogue gets past.
What the rules say about hiding
Stealing large chunks from my answer to What advantages does hiding have?
Best to go back to the source on this; bold italics are my emphasis.
PHB p.177
Stealth. Make a Dexterity (Stealth) check when you
attempt to conceal yourself from enemies, slink past
guards, slip away without being noticed, or sneak up on
someone without being seen or heard.
PHB p.177
Hiding
When you try to hide, make a Dexterity (Stealth) check. Until
you are discovered or you stop hiding, that check’s total is
contested by the Wisdom (Perception) check of any creature
that actively searches for signs of your presence.
You can’t hide from a creature that can see you, and if you
make noise (such as shouting a warning or knocking over a
vase), you give away your position. An invisible creature can’t
be seen, so it can always try to hide. Signs of its passage
might still be noticed, however, and it still has to stay quiet.
In combat, most creatures stay alert for signs of danger
all around, so if you come out of hiding and approach
a creature, it usually sees you. However, under certain
circumstances, the Dungeon Master might allow you to stay
hidden as you approach a creature that is distracted, allowing
you to gain advantage on an attack before you are seen.
Passive Perception. When you hide, there’s a chance
someone will notice you even if they aren't searching. To
determine whether such a creature notices you, the DM
compares your Dexterity (Stealth) check with that creature’s
passive Wisdom (Perception) score, which equals 10 + the
creature’s Wisdom modifier, as well as any other bonuses
or penalties. If the creature has advantage, add 5. For
disadvantage, subtract 5.
For example, if a 1st-level character (with a proficiency
bonus of +2) has a Wisdom of 15 (a +2 modifier) and
proficiency in Perception, he or she has a passive Wisdom
(Perception) of 14.
What Can You See? One of the main factors in determining
whether you can find a hidden creature or object is how well
you can see in an area, which might be lightly or heavily
obscured, as explained in chapter 8.
PHB Errata
The DM decides when circumstances are appropriate for hiding. Also, the question isn't whether a creature can see you when you are hiding. The question is whether it can see you clearly.
PHB p.183
A given area might be lightly or heavily obscured. In
a lightly obscured area, such as dim light, patchy fog,
or moderate foliage, creatures have disadvantage on
Wisdom (Perception) checks that rely on sight.
A heavily obscured area—such as darkness, opaque
fog, or dense foliage—blocks vision entirely. A creature
in a heavily obscured area effectively suffers from the
blinded condition (see appendix A).
Note: Blindsight is dim light and results in a lightly obscured area. Torches, lanterns etc are bright light.
Being seen is not enough for you to stop being hidden - you must be seen clearly. Lighting, cover and camouflage would all influence this.
Your specific questions
... can this Rogue sneak past with a disadvantaged stealth attempt?
Yes and furthermore I can see no reason why there would be
disadvantage to this. Anyone can hide where nobody can see them;
the purpose of the Dexterity (Stealth) skill is to be able to hide
where people can see you due to your skill, training and luck.
Do the orcs get a chance at Passive Perception to see the Rogue sneaking (using stealth skill)?
Yes - if the rogue's Dexterity (Stealth) check beats the orc's
passive Wisdom (Perception) then he sneaks past them.
Do we roll/check for the Sneak and the Orcs get to check Passive Perception?
Exactly.
The Rogue's speed is 25 feet (Dwarf/light armor), or 6 squares on the map and that gets him right near the orcs, but not past them, so
the orc should get a Passive Perception check before the Rogue gets
to move another 6 squares past them. Is this correct?
No, because:
25 feet is 5 squares on the map (just saying).
You do not use round by round movement unless you are in a combat
situation so if the rogue sneaks past then he sneaks past, if he
doesn't you may be rolling initiative and switching to round by
round movement then.
There is no passive [Wisdom] (Perception) check - it is a static
number; for an orc it is 10.
Effectively the rogue makes a Dexterity (Stealth) check against a DC of 15 (10 for orc's passive Wisdom (Perception) + 5 for advantage because there is more than one orc doing the looking and they therefore help one another), if the area is brightly lit and there is no cover.
If there are sufficient objects/people to hide behind or it is dimly lit (which includes darkvision) then the DC is 10 as the disadvantage for lightly obscured cancels the advantage from helping.
If the rogue is doing it in the dark (and he can see somehow) then it is automatic; barring making a noise.
Why not use active Wisdom (Perception)
Because the orcs are "standing guard" which to me does not engage "actively searches for signs of your presence." Standing guard is long, boring duty and you are not going to pay attention all the time especially when you have a couple of mates around to chat with.
If they were in a state of heightened alertness (like they knew you were out there right now!) then they could use active Wisdom (Perception).
Best Answer
Option 1: Answering the question as written
It's possible just to invert everything:
Option 2: The "triple agent"
Pick up the d20. Tell the player that you're rolling Perception for the guards, and you've come up with a scheme that gives the same probabilities of success and failure as if the player was rolling against the guards' passive Perception. Reiterate that you are rolling Perception for the guards, and not a Stealth check for the player.
Then, roll a Stealth check for the player.
Option 3: Schrödinger's check total
It's possible to prevent the knowledge of what total the player rolled without changing the overall probabilities. In fact, even the DM won't know what the player rolled. The only information that is produced is whether the player succeeded or failed against each guard, and not the exact roll. Here's how to do it:
Have the player roll against the first guard as normal. When the player encounters the next guard, have them roll Stealth again. However, if their total would be inconsistent with the result of the previous guard encounters, have them reroll the check. Keep rerolling until you achieve a total consistent with all past history. Then, forget the total again, remembering only the successes and failures.
Fancifully, you're travelling back in time and re-rolling the player's original roll in order to invalidate their knowledge of the total. However, you can't change an established success into failure or a failure into success. Or, the player's "real" check total exists only as a superposition of states with each "observation" (i.e. guard encounter) narrowing it down.
For example, let's say we have two guards at DC 10 and DC 15. Here are the possible sequences of events: