To answer your two questions in backwards order, but easier context:
Scene Framing
Splitting the party is easy and fun when you don't let scenes drag. Just as much as movies and TV cuts to relevant points, you should aim to start scenes as close to the important action as possible. Don't spend long on the set up, get to the interesting point of the scene ASAP. Throw clues in their faces. Put hard choices there too. And, cut away quickly too. Scenes should be 5-10 minutes at most. Since we're not talking high crunchy combat scenes, cutting away is very easy to to do here.
Known, Sought, Given Information
So, you're doing an investigation adventure, right? Go watch some investigation shows, read a few books. What happens in these stories? There is NO WAY for the protagonist to NEVER get the clues, it's really only a question of how they get the clues, how beat up they get along the way, and whether the clues come in time to do something about it or not. (Usual Suspects is a great movie example of getting all the clues too late.)
Information comes in 3 ways:
Known
Known information deals with things the characters ALREADY know. This can be inferred from their skills, their classes, backgrounds, history, etc. "You were a galley slave before being a pirate, you recognize the scars on the ankles from chains anywhere..."
Use this give each character plenty of context, plenty of "read" on characters or objects. ("He walks with a swagger, not the kind that comes with hardship, the kind that comes with having lived one's entire life at the top. You can see the difference anywhere. He's not one of you.")
Known information should often include lots of free clues or reads on things, because it gives players a feeling of expertise and competency for the characters. You can have dice rolls or whatever about specific questions or further clues, but start with the info their character can JUST SEE from the start.
Sought
Sought information has to be... sought out. This means it's not immediately obvious and either has to be collected ("pickpocket the letter tube from his bag"), or "processed" in some manner ("Scraping the iron shows it to be a softer type than normal. This was a cheap replacement, not the original.").
This is where character skills and player choices can be made, but since few players think of their characters as investigative types, you will want to provide some suggestions along the way. ("You've traveled far from home, but you're the best one on the ship when it comes to recognizing foreign plants. Maybe if you got a look into the doctor's herb bag you'd know what's in there...")
Also recognize that while a social character can con, trick, pressure characters into revealing information, the quiet high-perception character can often read other things about someone without directly interacting with them. Consider that a potential parallel method as well.
My suggestion is that if you have any kind of sought information available, make it something the players acquire/understand with just one skill check/dice roll/etc. Failure shouldn't mean "you don't learn anything" but it might mean "You get caught trying to get this", "The evidence gets destroyed/lost", "You only figure out what it means too late" etc.
Given
Here's a thing few rpgs get from investigative stories - a lot of clues just FALL into the protagonists' laps. People spill the beans, come forward, tell the dirt on someone else to get them in trouble, the heroes just happen to luckily be at the right place to overhear some incriminating statements, they stumble upon a crucial clue left forgotten at a crime scene... this stuff happens a lot. The only reason other media gets away with it is that the heroes often suffer so much it's like "well, sometimes you gotta get lucky, right?".
One of the better rpgs to deal with investigation is Dogs in the Vineyard, which has a pretty simple bit of advice - have several characters try to GIVE the information straight to the PCs... lying or omitting just enough to cover their own asses or their friends. The other bit of advice is that straight out lying should have the GM say to the players, "You can tell they're lying, you're just not sure what the real truth is."
AS long as everyone is at least looking for information, one of these three types should be available to give clues or at least ideas on how the characters treat/feel about each other.
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.
Best Answer
I used to play a game that was fun and exciting: you rolled a dice and depending on the result moved up some ladders or slid down some snakes and the first one to the top won, its name escapes me for the moment. It was thrilling and intriguing and then I turned 5 and realized it was no fun at all because I had no agency.
My definition of agency is:
To qualify as a informed decision there has to be:
Please note that nowhere in my definition did I mention anyone rolling dice. Dice are only needed when there is uncertainty in the outcome: choices do not have to have uncertainty.
With this in mind let's look at your real life examples:
The blocked trail
This clearly qualifies as a decision, there are three choices here - two obvious and one hidden - that all lead towards the goal and all with different risk/reward/cost profiles:
There is a clear trade off here: time vs effort. Neither of these translate well into an RPG context because its the characters who pay the cost not the players but let's see what we can do to overcome this.
We have to put the cost in terms that the players will care about so lets assume that there is a time limit on the adventure: they have to get somewhere by a certain time or something really bad happens - the evil cult summons the demon, the corporation finishes their hack, whatever.
So, the players are informed of the costs of the two obvious options and they matter in game terms: the trade off is reduced capability in future encounters versus definite lost time. In addition, you have indicated that they can do something else - if they spend 10 minutes scouting they will find the path or the could use up other resources like a spell or come up with something I haven't thought of.
Note that only 1 of the options (pushing through) has an uncertain outcome so only if the players choose that one does anyone roll a die.
The point is the players now have agency - a decision with foreseeable consequences.
The flooded creek
This one is even easier as all the choices have in-game consequences - the risk of being swept off the waterfall versus splitting the party.
You could put in a saving throw here if you really want to to see if she makes it.
You don't need to lay out options here because they are pretty obvious - go forward and risk drowning, go backwards and risk drowning, split the party or use up some limited resources. You can work out the appropriate game mechanics and tell them to the players so they know the risks, only if they try to ford the stream are dice needed.
Again you have given the players agency.