Erina Nakiri has looked down on Soma Yukihira's cooking because he comes from a plebeian background, and doesn't believe in using the highest end ingredients in his food (among other reasons). Though she's not the only person with this mindset, she embodies it the strongest, and a lot of her frustrations regarding Soma revolve around this. With the introduction of her father Azami Nakiri, it's clear why this is a sore point for her - Azami belittles, embarrasses and insults his daughter and her customers, claiming any who believes high quality ingredients equates to high cuisine is an idiot who shouldn't be served food in the first place. In other words, he's the extremist example of this viewpoint.
In this same line, she tends to mistake fear for respect, and it goes both ways. Her upbringing has made her afraid of her father, yet she nonethelss respects him despite the abuse he submitted her to. In turn, she's so used to people being intimidated in her presence that she takes Soma's irreverent attitude as disrespecting her, only because he doesn't let her intimidate him.
In Chapter 3, Erina fails Soma on his entrance exam despite her clearly enjoying the dish he served; Soma returns in the following chapter showing his application was accepted as her decision was overruled by Senzaemon Nakiri himself. It might seem odd the director of a school as prestigious as Tootsuki would take interest in a single student, even if the quality of his dish is good. In Chapter 135, Senzaemon confessed he advised Jouichirou Saiba to have Soma apply to Tootsuki; not only did he deliberately ensure Soma would be brought into the student body, but Senzaemon agrees with Jouichirou that his son is the one who can change Erina for the better.
A bit of Fridge Brilliance mixed with Shown Their Work: When Megumi was seen butchering an monkfish, a lot of students were shocked that someone as shy as her is able to do the intricate butchering steps to break down a monkfish. If you know anything about monkfish butchering and the sheer amount of training that goes into it, not only is it really hard to do, but most people who practice it are often trained at a very young age. In addition, most people who are trained in the technique tend to be boys, and even then, it is usually only the oldest son in the family because of how expensive monkfish are. So it's not just the fact that Megumi knows how to pull off such a technique that gets the interest of the crowd, it's also the fact that Megumi is a girl that knows the technique that gets the interest of the audience.
While Erina naturally does expect her number two to cook well, Hisako never gets looked down by her for her losses. That's because as Hisako's specialty is medicinal cooking, her food first and foremost is supposed to make the eater healthier. Actually tasting good is merely a bonus, and, as such, Hisako's food can't be criticized.
The Stagiaire Period Arc and Callbacks to it during later Story Arcs are full of this. It showcases Totsuki's commitment to forcing its students to improve by going far above and beyond what most lesser institutions would consider "good enough".
From the start, it's shown that students are required to do far more than just handle their assigned work without any problems: they are required to make lasting contributions to the establishments where they work. Furthermore, said contributions need not necessarily be directly related to the cooking itself, as Soma and Hisako demonstrate when they help their first workplace solve the problems caused by the change in its environment.
It is demonstrated that the faculty are seeking to actively push the students to excel by forcibly removing them from their comfort zones, to work in areas where they are not as competent or comfortable - Megumi Tadakoro is sent to a high-class French restaurant, run by an incredibly domineering chef, a difficult environment for someone as shy and retiring as her, where she has to assert herself to make her suggestions heard. Similarly, Soma is sent to Koujirou Shinomiya's restaurant, another French restaurant, a major departure from a diner setting he is used to doing and has to adapt and learn on a very steep curve in order to succeed. Finally, a Flashback during the Elite Ten Invasion Arc reveals that Ryou Kurokiba was sent to an Indian restaurant where he was forced to contend with extensive use of spices and seasonings, having to learn about them in order to succeed. All three examples eventually leads into...
The ultimate aim of the Stagiaire sessions is not just for students to experience Real Life work situations, but also an opportunity for them to see and work with professional chefs and restaurant owners up-close and personal, giving them the opportunity to broaden their repertoires by "stealing" skills from those chefs to make their own.
Rentarou Kusunoki's Rummage Sale Reject of a school uniform may seem over-the-top at first, but once his Inferiority Superiority Complex comes into context, it makes perfect sense: he's actively trying to stand out from the crowd, in any way possible, and ends up overdoing it as a consequence.
It's no wonder that Soma was the only one of the "Rebels" who passed the third test in the Hokkaido Promotion Exam Arc against a member of the Council of Ten Masters: Soma had the support and direct help of an entire research society; furthermore, his opponent was off his game and refused to use any resources his position on the Council granted him. Meanwhile, even with Erina's training, the rest of the Rebels (sans Megumi and Takumi Aldini) wouldn't have done better for the following reasons:
Shun Ibusaki, Ryoko Sakaki, and Yuuki Yoshino specialize in preparing time-consuming ingredients that they likely didn't have access to; likewise, Zenji Marui was cut off from his books, which requires time and research.
Daigo Aoki and Shoji Sato work best as a team: given that Central was splitting the Rebels up individually, one can assume they were tested one at a time against a Council member.
Meat-specialist Ikumi Mito was facing sweets specialist Momo Akanagakuro; by default, the odds would be stacked against Ikumi if the theme of the ingredient was something that clashes with meat, which it likely was, as Central was already directly rigging the examinations.
Ryou was limited to what seafood he could use, since most of those products are out of season during the winter, which doubles as a Call-Back to the first test involving salmon when the Story Arc began.
Alice Nakiri, who's taking a stand against her Evil Uncle, probably was cut off from high-end gastronomical machines or no machines to use at all.
Poor Isami Aldini was straight up competing against the First Seat.
It's shown that Azami deliberately sought out the members of the Council of Ten Masters that voted him in and swayed them to his side by appealing to a specific facet of their past and/or personality: Eishi Tsukasa's perfectionism, Momo Akanagakuro's elitism, Soumei Saitou's nobility and devotion to his ill mother, Nene Kinotuni's traditionalist mindset, and Etsuya Eizan's greed. The exception seemed to be Rindou Kobayashi, whose stated reason was For the Evulz (though it's implied, however, her actual reason was Undying Loyalty to her best friend Eishi).
By contrast, the members who didn't vote for Azami, the ones he never bothered to meet with, have legitimate reasons to oppose Central's ideology:
Tosuke Megashima's goal is to help out and protect the ramen chefs of Japan, thus standardizing a way of making ramen jeopardizes that.
Satoshi Isshiki values innovation over tradition and encourages this mindset in his friends and underclassmen.
Terunori Kuga is the head of the Chinese Research Society, which Azami's reforms required he dissolve.
Like Rindou, Erina is the exception in this group: while she agrees with her father's ideals to a degree, those thoughts were completely overshadowed by his horrific abuse of her during her childhood — she'd never vote him in simply based on that alone.
Soma challenging Erina to a Shokugeki within the Régiment de Cuisine finale seems out of place and redundant: in order to beat Eishi and Rindou for the win, she has to create an entree that surpasses Soma's appetizer. However, within context of Erina's Character Development, this little roadblock makes sense. If Erina's dish is superior, it's because she threw out everything she knews about traditional cooking by coming up with her own cooking style, thereby invalidating Azami's ideals; if she can't surpass Soma's appetizer, that means he's basically beaten her in an indirect Shokugeki, proving he's the better chef (incidentally, this also gives Soma a Heads I Win, Tails You Lose situation against Erina).
Asahi is the name of a Japanese beer. In Chapter 289, we learn that Asahi's mother was an alcoholic. Who's to say she didn't name her son after the alcohol she drank?
Why is the graduation rate so low at Tōtsuki? The same reason why at least 60% of restaurants fail within 3 years. Tōtsuki wants to guarantee that any graduate has the ability to make it running their own restaurant, which requires not just the ability to cook, but also flexibility, diversity, and networking amongst other skills. If a student fails to break out of the "Good cook = good chef" mindset, they wouldn't have succeeded in the real world anyways. The competitive nature of the Shokugeki also falls in this mindset. If/When you have to compete against another restaurant, simply making good food isn't enough (as anybody who watches Food Network knows). Having the ability and pride to make something of a "specialty" that maximizes flavor along with presentation and uniqueness is what would make your restaurant stand out in the real world and keep it in business. The losing "restaurant" has to face serious consequences from losing "customers" (lack of budget, network connections, etc.) to even closing the business (expulsion). The expulsion rate being higher than the actual restaurant failure rate is Tōtsuki's way of ensuring every student will be a success, rather than passing potential maybes.
To reinforce this fact, graduates go on to either revolutionize the cooking world or go on to be leaders in the field, even Megumi for how specialized her Comfort Food can be can find herself as head chef of an inn where her hospitality based cooking style is essential. Even if they branch away from the kitchen, their skill sets can allow them to support restaurants or the cooking world as a whole considering the innovations that can be done outside of the kitchen. In other words, if your aim isn't to be the best in the cooking world and have no desire to go beyond that, then you shouldn't be at Tōtsuki.
During the training camp arc, there's another good reason that eggs would be assigned as the main ingredient of choice besides it being a popular breakfast food and very diverse to allow for multiple recipes. Even though the students all tended to make smaller dishes for tasting purposes, eggs are a food that are very high in protein. Which, consequently, means they would fill your stomach fairly quickly depending on how much you ate. Even with the massive amounts of people coming to eat, the majority might only be able to try two or three dishes If they are the whole thing instead of just having a taste. Not only is there a time limit, there's a hunger limit. If you couldn't attract customers to your dish by making it look good and not just taste good, all the customers got full before they could ever try yours. In addition, as any good chef will tell you, egg-based foods are the second worst thing to serve in a buffet, right before fried food, which can use eggs as an ingredient, because eggs absorb a lot of moisture from the air, doubly so when fried, and it's even demonstrated when Soma finds out the hard way.
Azami apparently was a chef good enough to satisfy Mana's God Tongue, but only for a while. Given that he neither has the drive nor sees the need to improve his own cooking (unlike Joichiro), it's no wonder that he never sought out new flavors his wife could enjoy.
Mimasaka's default strategy would have failed utterly if he'd been matched with the wrong opponent during Spring Selection. He wouldn't be able to copy Alice's dishes without her array of expensive machines, nor Hayama's without his supernaturally sharp nose and spices that grow nowhere else in Japan.
Erina should be grateful that she met Soma, because she wouldn't have otherwise become as great as a chef. Think about it: it's all but stated that she, like her father, coasted on her natural talents to breeze through everything, but even natural talent can only take you so far. At the start of the story she's so feared that there's almost nobody who dares challenge her, so she had little to no incentive to better herself. Meanwhile, Soma has competed against his father for his entire life, and while he's never won once, he doesn't let that deter him from trying to go beyond and create new dishes, and while he acknowledges Erina as the better cook, he still continues to challenge her and improves every time. Had Erina stuck to Azami's mindset, Soma would have long caught up to her and even perhaps surpassed her, so she realized she had to work hard like Soma and try new things so he wouldn't catch up to her. Natural talent cannot compete with natural talent and hard work.
. In Chapter 289, we learn that Asahi's mother was an alcoholic. Who's to say she didn't name her son after the alcohol she drank?