No Face is a monster who takes on the emotions of those around him (he's helpful to Chihiro because she helps him, etc). It only occurred later that that's why he's called No Face—he doesn't have any personality of his own.
It also doubles as a pun. His name isn't just No Face he also had a Noh face!
In the Japanese version (with English subtitles), Yubaba observes that the "Stink Spirit" choose to hide itself in the rain. So, of course it would be so appropriate for the River-Kami to select a rainy day that might quell his stench and pain and allow himself to be accepted (a little more easily) into the bathhouse. It could also be that, as a water spirit, rain follows him everywhere, or he is at least capable of summoning it at will.
It is quite appropriate for Chihiro to be the first to notice the pollution lodged in the River-Spirit and pull out the last speck of pollution (a fishing hook). After all, humans polluted this River, so Chihiro represents the potential of humanity or/and atonement for human's problems.
In the middle of the story, after Chihiro gives the River-Kami a bath, he flies away and we can see him after being cleaned up. It's very fast, but it's possible to figure out he's a very elder and white dragon. And who else is a white dragon? And what does that person turn out to be in the end?
The only thing that can cure Haku is the River Kami's gift. It's logical that medicine from a river would cure another river.
Chihiro tries the medicine not long after being given it, only for it to make her gag with it implied to be quite disgusting, as is common with medicine. However, both times it is used, saving Haku from Zeniba's curse and making No Face release his victims, it is precisely the fact that it induced vomiting that made it useful.
It's also worth noting that the medicine's ability to purge or cleanse mirrors how Chihiro helped cleaning the river spirit from things that damaged it.
Adding on to that, it's mentioned that the river spirit Chihiro helps cleanse of pollution is very powerful, and that Yubaba had to put a black slug into Haku just to control him, whereas stealing names is enough for everyone else. It stands to reason that river spirits would be quite powerful even without this evidence (rivers can be rather formidable forces of nature), so it's possible that Haku doesn't react because, free from the slug's influence, he's actually more powerful than Yubaba.
When Chihiro chose that neither of her parents were in the pig herd presented. That scene also has another meaning that basically says that her parents are not pigs for she is a human being, not just a mere animal.
In the Japanese version, it's easy to tell what's going on if you're paying close enough attention. Chihiro is able to figure out with very little information that the dragon she observes getting attacked by paper dolls outside the bath house is actually Haku, and she expresses surprise that Yubaba can't recognize her son when he's traveling with her in the form of a mouse. Chihiro seems to have a talent for seeing into the heart of a person. This is exemplified when she looks into the pen full of pigs at the end of the movie and realizes none of them are her transformed parents.
If Yubaba and Zeniba are truly the same person, it would add a new dimension to the scene where Zeniba tells Chihiro she has a pretty name and reminds her to take care of it - this could be Yubaba withholding ownership of what she took from Chihiro earlier. (In the Japanese version, she clearly says, "Take care of [your name]. It's yours.")
The reason Chihiro didn't turn into a pig like her parents is because she only ate food that was offered to her by a spirit. Fridge Brilliance comes that the only times Chihiro is seen eating in the spirit world was when someone offered her food. Two times were by Haku (what appears to be a fruit and rice balls), another time by the River Kami with the medicine, another by Lin with the pork bun and at the end with Zeniba with the tea and desserts.
The fact that at no point in the movie did any of the characters, aside from her parents, treat Chihiro as if she were a child. In the Spirit World, she is just another human, regardless of her age. She's expected to pull her own weight and do as well as any of the other workers in the bathhouse, which is one of the factors contributing to her maturity throughout the course of the movie.
At the beginning, everything Yubaba says about her is true (a spoiled, lazy, selfish girl who has no manners), and towards the end, the workers have begun to see her as an equal. It's just another extremely well-thought-out indication as to how far she's come in her emotional maturity. She's forced to say yes ma'am and no sir, and do what she's told without question. Spirit journey aside, this is part of the reason why her character development went as far as it did.
After Haku is injured, Chihiro tries to go up to the penthouse to find him, but is stopped by one of the bath house attendants who freaks out and lets go of her when he sees that her hand is covered in Haku's blood and some of it has gotten on him. This makes sense as he's a Shinto kami, and in Shinto, blood causes spiritual or ritual impurity, a rather large failing for a bath servant.
Haku is the very first person who's friendly to Chihiro and patient with her fear and loneliness. On finding out his real identity as the Kohaku River, of course this makes sense. Humans gravitate to bodies of fresh water for survival, so he's probably lived for centuries with them.
It seems odd at first that Haku tells Chihiro to get a job from Kamaji when he ought to know that Kamaji employs sapient sootballs to do the work for him. In fact, Haku could have easily taken Chihiro to Yubaba to get her a job, without all the hiding. But there could be possible reasons for Haku to want Chihiro to seek out Kamaji's help:
1) Chihiro needs friends in the bathhouse. She needs other beings to look out for her when Haku can't, as shown when Kamaji covers for Chihiro when Lin brings him lunch, and when Lin despite herself starts to care for the younger girl. Haku has to spend his day and night running various errands for Yubaba, while Chihiro would be working in the bathhouse.
2) Yubaba could have easily realized that Chihiro was special to Haku, and might have stopped the duo before Haku and Chihiro could even ask for a job. It wouldn't have been hard for her to compel Haku to even kill Chihiro or stop her in such a fashion.
When the river god/stink spirit passes by Lin holding two bowls of rice, the rice turns black and crumbles. Given Lin is cringing from the smell, it might be safe to assume that it was implying it stank so badly that it spoiled the rice. However, rice is also known as a sponge for negative energy and impurities. It wasn't that the spirit stank that badly, the rice was simply absorbing the impurities from the polluted river spirit.
When Chihiro comes to the aid of the bath house staff, and Yubaba comes to confront her, she sees the mouse on Chihiro's shoulder (actually her son in disguise) and is vocally disgusted, along with believing the three heads transformed weren't her actual son. This may be because he was disguised with her sister's magic, but seeing as Yubaba has similar magic she should have seen through the trick. This is brilliant in its own way, as it sets up Chihiro eventually seeing through the trick of Yubaba's final test. Chihiro has the heart to see through to what other peoples' true selves are, while Yubaba is sharp and believes only in what she sees outwardly of people.
One of the more innocuous moments can become truly horrific: Yubaba threatened to turn Chihiro into a piglet or a piece of coal, implying that these are standard punishments for intruders in her realm. Now, think of the sootballs carrying pieces of coal to the roaring furnaces by the hundreds, and Chihiro briefly helping them with this job.
There's also the fact that Chihiro would have been eating solid meals the entire time she was at the bath house — those porkbuns? PEOPLE. Let's hope that the spirits also breed normal pigs in addition to taking polymorphed humans. After all, there couldn't be that many people getting themselves caught in the spirit world... right?
Even if the coals are transformed humans, the horror of it is lessened slightly when you realize that transformed creatures dont remember they were originally anything else. So at least the humans, being turned into non-sentient objects, wont feel a thing when theyre tossed into the fire.
The train may be the main mode of transportation for souls, but clearly it goes to more places, as evidenced by Chihiros ride to Zenebas house; this proves that its not just a vehicle for the dead. Lin likely wants to ride it so she can explore the rest of the spirit world, not because she wants to die.
The fact that Kamaji tells Chihiro that the train only goes one-way. First, you might only think about how physically she wouldn't be able to return to the bathhouse by train. Then you realize that this is an Afterlife Express and Kamaji mentions that the train used to go both ways. Those that go on this train (aside from Chihiro) depart from the spirit world entirely.
At least one of these faceless spirits is shown to be a little girl standing on a platform as the train moves on to its next destination...
Look carefully at the tunnel right before Chihiro enters the spirit world, and right after she exits. In the time she spent there, the arc has become overgrown with moss, the face of the statue has completely eroded away, and the car has notably become covered in leaves. While the car starting up means that they couldn't have been gone for too long, Chihiro's parents are probably in for an unpleasant surprise when they try to go to the house they were supposed to move into that day (and which Chihiro's mother was worried about them being late for,) not to mention her father's new job that would have been expecting him.
Its not explained what the black shadows on the train are. However, its possible that theyre the souls of the humans that have been killed in the spirit world over the years. It would explain why theyre on a train in the spirit world: As soon as the humans bodies die, their souls are released, and they take the train back to the human world so that they can move on to the next life.
When Haku brings Chihiro to see her parents, you can briefly see a piglet standing in a pen among the adult pigs. Later, Chihiro sees the shadow ghost of a little girl on the trains platform. That little girl was probably the piglet, killed in the kitchen and then returning in ghost form to get on the train. It can be assumed that her parents havent been killed and eaten yet, which is why shes all alone on the platform: shes waiting for them to join her.
When Lin tells Sen that Haku is Yubaba's henchman and not to trust anything he says, Sen immediately crouches down and sobs. Maybe Sen thought she'd been conned by Haku while vulnerable, into going to Yubaba and giving up her name, therefore forced to a life working in the bathhouse? No wonder she was crying all through the night.
Haku was sent to steal Zeneba's gold seal. In most legalities a seal on a will is just as legal as a signature. Since Yubaba can only control those whose names she's stolen, she wanted that seal to control her sister. No wonder Zeniba was mad.
If Sen and Haku had never met as Chihiro and Kohaku, the river spirit, she would have probably been long turned into food, her parents forever pigs, and Haku would have remained a slave to Yubaba .... forever.
By "Don't look back," Haku also meant that Chihiro should not tell anyone else about her experience in the spirit world after she returned.
It may also be a reference to the Greek tale of Orpheus - he went to Hades to save his lover, Eurydice, on the condition that he doesn't look back until he leaves. He does look back when he is out, but not when Eurydice leaves - leaving her trapped in Hades for eternity.
Or Lot's family in the Bible. Don't Look Back is a common mythological/folkloric trope across many cultures, one of many that Miyazaki channels into his works.
Eating food at an apparently abandoned theme park is a pretty bad idea, even without spirit world creepiness. They just start eating food that has just been sitting out for— if not years, months, weeks, or days, then at least hours.
While it was a superbly stupid idea to assume that all that food lying there was okay to eat even if they intended to pay as soon as any personnel showed, the food did smell and look fresh out of the oven/pan/pot/whatever that respective dish was made in and despite it being warm enough weather to not wear sweaters or coats there were no flies flying around.
It's worth noting that the act of simply smelling the food may have started affecting the parents' minds almost immediately - as soon as they smell it, it's all they care about. Granted, while they had been traveling for likely several hours and were probably hungry and wanted to stretch their legs, there's still a few additional cues:
Chihiro's mother impatiently tells her to "hurry it up" as Chihiro struggles to cross the river with its huge rocks. Neither parent really makes an effort to help her, but the dad actively extends a hand to the mother to grab. They begin heading off without her. Bratty or not, parents typically keep their kids close in unfamiliar areas.
Even if they had intended to pay for the food as soon as the workers came back, they seem to almost immediately attack the food upon reaching the stalls with little to no hesitation. Sure, they were hungry, but typically, even hungry people at least converse during their meals, or start to eat a bit slower. Chihiro's parents almost immediately started GORGING themselves once they'd had that first taste. They barely slow down to consider what they're eating and eventually start to take absolutely no notice of Chihiro.
Before Chihiro walks away in frustration, and even before they've even eaten a lot of food, you can hear very loud snorting and grunting from both of them as they continue to eat. Humans do not typically make these types of noises while eating. It's likely that they started to turn almost immediately after that first bite of food.