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.
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.
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.
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.
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?
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?
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).