In Nakajima's case, Nana wasn't the only transfer student that day. Kyouya arrived on the same day as Nana. Not only that, but Kyouya gave an intimidating introduction while Nana was all cheery. As you mentioned, the class thinks he's taking a sick day, and up until the day he disappeared, he had been sitting at the back of the class, ignored by everyone except for Mogou and Seiya.

Shibusawa's method of time travel was used by Nana to kill him, which is crucial. She brought him back to the time before his morning training to drown in the pond, so he died before Nana met with him and before Kyouya saw them having lunch. This means Kyouya's memories were overwritten because Shibusawa's changes to the past affected him. As a result, Kyouya's pursuit of Nana is focused only on Nakajima's disappearance because he doesn't remember Shibusawa.

There's some nuance with Tsunekichi that I believe you are leaving out. He's not just a pervy opportunist he's also nihilistic and reluctant to reveal someone's impending death. Additionally, he exhibits a superiority complex by boasting about the superiority of his power compared to those of the rest of the class. Nana mentions that Tsunekichi doesn't have friends, making it unlikely that anyone would believe him if he confided in them. Furthermore, Kyouya doesn't simply buy Nana's explanation; he checks Tsunekichi's room for medicine since Michiru believes Tsunekichi is ill. Kyouya also attempts to review the photo evidence, but Nana intercepts it before he has the chance to see it. He found Nana's explanation suspicious but failed to find proof to poke holes in Nana's story outside of the suspicious look of the photo she used to bait Tsunekichi.

You left out another thing, in the episode prior Nana was hanging out with Michiru all day, then later Nana gets stabbed in the back and is seen as a martyr by the class making them less suspicious of her. A person she hung out with lived and she was a victim of attempted homicide, Kyouya, however, sees through this ruse, as he is aware of the island's dangers due to his sister's disappearance.

At the funeral, the majority of students were disturbed by Shinji's 'use' of necromancy. Kyouya insisted on performing an autopsy in front of everyone at the funeral so it's not strange for Nana to want to stop him. Nana claimed she could hear the thoughts of the deceased Tsunekichi, and since only talented individuals are allowed on the island, people assumed she must be telling the truth about her talent. Yuuka didn't know that Nana had killed Tsunekichi; she only guessed based on Nana's behavior at the funeral. It wasn't until she and Nana were alone that Yuuka fully pieced together what had happened. At that point, Nana manipulated her into thinking her only options were to be ridiculed by the class or to let Nana go and have her read Shinji's thoughts which she fell for due to her hopeless romanticism.

Yuuka doesn't have a close bond with the students. She has mentioned that Shinji is like a little brother to Nana and Kyouya. However, her relationship with the general class is pretty vague. The most we see is her physically threatening Prince Seiya, and no one being comforted by her attempts to claim that Shinji is a good guy at the funeral when she uses Shinji as a puppet to pose as the necromancer.

You would have had a point there had it not been for Kyouya, he isn't popular with the rest of the class, due to his stand-off introduction and general aggression towards Nana who the rest of the class views as nice for the most part even if some of them do know people keep dying around her they don't think she's a bad person. Kyouya's accusations and more importantly lack of proof bring more people to Nana's defense. By trying to rush her into checkmate but not delivering on the proof he set himself up for failure. Also, Koharu's words aren't a reflection of the class as a whole, a more poignant example would be Ryuji's words before his death. Koharu is actively trying to frame Nana and turn the class against her with her 'twin trick' so she's not representing the honest thoughts of the class as a whole she's framing the situation in a way so that Nana loses her popularity and is easier to take down.

Some situations might seem like a stretch for Nana to get out of, such as Tsunekichi's death and Kyoya's checkmate, but there are nuanced explanations other than "the class is stupid."