The students of Class 1 - A are often involved. For instance, Midoriya might be portrayed as being unjustly treated by Aizawa, and the fanfic could focus on Midoriya's struggle against Aizawa's supposed unfairness. Bakugo could also be in the story, maybe being in conflict with Aizawa's disciplinary actions.
Aizawa is still going strong in the manga. There's no event or storyline that has resulted in his death. The creators have chosen to keep him as an active character.
Often, there is the theme of inspiration. Deku is very inspiring in the original story, and in fanfiction, he can inspire Shoto to be a better hero. Also, self - discovery. They both are on a journey to discover more about themselves, their powers, and their place in the hero world. In fanfiction, this journey can be elaborated on in different and interesting ways.
One common theme is redemption. Dabi, with his troubled past, often has a journey towards redemption, and Shoto can play a part in that. Another theme is family. Given their complex family relationships in the original story, fanfics often explore how they deal with family issues together.
No, it is not appropriate. Creating or distributing sexual fanfiction involving underage or fictional characters in a sexual context is unethical and often violates the terms of use of various platforms. Such content can also be considered a form of inappropriate objectification.