Stealth is fun.
Shadowdancer may be one of the most popular Prestige Classes in 3.5e, and that is solely due to the Hide in Plain Sight feat. Many players enjoy the thought of sneaking invisibly to the enemy and rolling insane backstab/sneak attack damage.
Unfortunately, stealth in D&D is not always that fun.
Now, the backstab part is awesome, and that's why most stealth players enjoy it. The problem is that the D&D mechanics as they are played out in most campaigns do not make much of stealth beyond a canned skill challenge. By looking at some good stealth games for the computer, such as Dishonored or Assassin's Creed, we can take some tips and add them to our campaigns.
Objectives
This is the biggest change that a DM has to foster in his campaign. As mentioned before, the objective of stealth is almost always just to get in some extra sneak attack damage. Stealth gets boring when, in the end, it's only about combat. There is nothing wrong with sneak attacks, of course. Some of the most memorable moments in my campaigns have been sneak attacks (double crit + 4x backstab damage FTW?), but stealth needs variety.
The purpose of stealth is to remain undetected. Let stealth be a tool for defeating encounters. If the players successfully sneak around an entire group of hobgoblins, give them full XP as if they had beaten the fight. And don't just stop there. If you want great stealth encounters, turn it into a real challenge like Dishonored does. Make enemies move around somewhat unpredictably. Have your players use distractions, or find opportunities to pick off the enemy one at a time. Give them bonus XP or a better reputation for being able to complete encounters without bloodshed, similar to Dishonored. Also like Dishonored, make a few combat encounters really dangerous if you rush right into them, and be sure to make that fairly clear through in-game information.
Environment
The world is bigger than a grid. Description helps. Open up the terrain for movement, like Assassin's Creed. Let them sneak past the royal guard by balancing across the rafters of the great hall or by sneaking over the rooftops to bypass the thugs waiting for them in the street. Think in 3D even though the grid is 2D.
Light is a huge factor for stealth in a lot of games, such as Amnesia: The Dark Descent. It ought to be very important in D&D as well, what with all the torches, lanterns, and magical lights often found in its environments. Have players make strategic use of light. One campaign, my players doused a torch while the guard was on the other side of the building so that when he came back, he couldn't see them sneaking inside. Unfortunately, the sudden lack of light alarmed him, which leads to another point:
The Chase Sequence
The way you describe the ninja character as cycling through backstab -> run -> hide -> backstab definitely confirms this as a bad pattern of stealth. One of the biggest flaws of the first Assassin's Creed game was how you could stab someone, run like heck, hide on a bench right around the corner, then go back and stab someone else. Rinse. Repeat. Worst of all, until they introduced notoriety in later games, it seemed like everyone forgot what you did.
Dishonored is a much better example of how to do detection and chase effectively. On the very first detection, the enemy is immediately alert and aware of the fact that you do not belong here. Hostility begins right away, and the chase is brutal. In a chase, NPCs do not let you get away unless you do something really daring. In the TV Show Burn Notice, the main character remarks during a narrative in a chase sequence that the only way to escape a chase is to do something that the people chasing you won't do -- like jumping off a roof.
And even if you get away, the NPCs should not just "forget" about you. They should be on high alert until you die or they are convinced that you have been driven off. Enemies on high alert for a stealth PC should not be easy to catch off-guard. In addition, they should not be splitting up alone if they are even reasonably intelligent.
Have NPCs adopt tactics like the PCs tend to act when encountering stealthy foes.
All Alone
"But I'm the only stealth character on the team!"
This is roughly the ninja player's position, I take it. I've been there. Fortunately, you don't have to be reliant on stab-and-run to be useful. A number of the former tips are intended for stealth-based encounters, but here's what a stealth PC has gotta do to have fun with stealth while your allies are kickin' down doors in the name of Tempus:
1: Wait for the encounter to get started. Be out of sight on the periphery.
2: Sneak up to a squishy target.
3: Stab.
4: ???
5: Profit.
An ultra-stealthy character is ideal for taking out priority targets. Then, using other skills, such as acrobatic-type skills, make a daring escape. Not just running away by pure movement points, but dodging between pillars, leaping onto ledges, or tumbling past enemies to rejoin your allies.
Stealth should get you into the fray. Speed and tricks should get you out.
First, this adds variety to your actions as a stealthy character.
Second, it should be hard to lose detection when enemies are tracking your movements so closely.
Beyond combat, a stealthy character can still be a great asset. Perhaps you can open a gate while the party is fighting. Maybe you can sneak into a camp and rescue a prisoner while the party is attacking from the opposite side. Generally, you should avoid going too lone wolf unless your party wants you to do so, because that's dangerous and slows down the game for others. Performing a stealth mission while the party fights a battle has been the best possible scenario in campaigns I have played. It keeps everyone busy, provides a distraction, and lets your group benefit from stealth simultaneously.
TL;DR version: Stealth should be more than dice rolls. It needs to interact with the environment and the intelligence of the NPCs involved. It should be rewarding, fast-paced, and require cleverness more than just sneakiness. When done right, it should give the party big advantages as a whole.
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.
Best Answer
There are several options, some more devastating than others.
I'll list my three favorite ones here:
Chase the Rogue
As Guildsbounty mentioned, a hidden Rogue isn't invisible: it just means their enemies have lost track of where they are. But their enemies still know where they were. You can run behind the last object you saw the Rogue go behind. Unless the rogue is moving particularly quickly, the Rogue is likely now visible. You should note that even while visible, the rogue may still be hidden. However, they at least will no longer be "unseen", which means they will not gain advantage on an attack roll made against their pursuer (which is crucial: more on that later).
Ready Actions
A rogue will give away their possition when they attack. So you can ready an action to injure the rogue when you locate him or her. This tactic is somewhat flawed for a couple of reasons. For one thing, you'll usually need to be attacked before you take your held action, which could mean you'll take considerable damage. For another thing, if you're a martial class you'll lose access to your Extra Attacks, and if you're a spellcaster you'll need to maintain concentration on your readied spell (particularly problematic if you're going to be attacked before you cast the spell). But still, it will permit you to do some damage to the rogue whenever it attacks you. This leads me to a third and crucial tactic.
Give the Rogue Disadvantage on their attack
There are several ways to do this: you could become invisible yourself (or simply obscure your location through a spell like Darkness or Fog Cloud), or restrain the rogue somehow (like through the Entangle spell) or simply go prone if the rogue is using ranged attacks (and can't walk up to you safely). Whatever method you chose, you need to ensure that the rogue doesn't have advantage when attacking you.
Note that the rules on advantage and disadvantage state:
Since advantage and disadvantage cancel (no matter how many times each is applied), that means that if you somehow do something which imposes disadvantage on the rogue's attacks, the rogue cannot then gain advantage (no matter what other sources of advantage they have). The rogue will be attacking you without advantage, and (since this is PvP, presumably) without another enemy of yours within 5' of you. That means that you've removed one of their most essential features: their Sneak Attack. Without it, a rogue's damage output plummets. They're essentially a martial class with a single attack per round, and lower hit points.
A spellcaster who simply goes prone and readies a fireball (or more likely "shatter" to circumvent the rogue's Evasion feature) every turn will quickly see the rogue's tactic fail spectacularly. A martial character may have a more difficult time, but they will likely find themselves dealing as much damage as the rogue every round if they ready actions while prone (again, make sure the rogue can't reach you), while sporting higher hit points.
No class is designed to be completely independent: we all rely on the support of our fellow party members in ways we may not even realize. A rogue, without companions to distract enemies or give them sneak attack opportunities, may find themselves acutely aware of how much their class is not designed to fight a canny enemy alone.
NOTE: Sneak attack does not require the Rogue to have an ally within 5' of their target, but rather an enemy of the target to be within 5' of it. So in a massive PvP arena, it's possible the rogue could still target a person who is imposing disadvantage on attack rolls, and still gain sneak attack. Particulars of the battle (how far away you can get from other characters, for example) will make this more or less effective.