The Neon Genesis Evangelion manga has indeed finished. It reached its conclusion after a series of chapters and story arcs that wrapped up the plotlines.
Sure! Neon Genesis Evangelion has a manga version. It presents the story and characters in a different format from the anime, but still retains the core elements that made the series popular.
No way. Canonical content in Neon Genesis Evangelion is what was produced by the official studio and creators. Fanfictions, no matter how interesting or popular, are not part of the official lore. 'Lemon' fanfictions, which usually deal with more adult - like or unorthodox relationships between the characters, are completely fan - created and not recognized as official in any way. They are a form of fan creativity, but they have no bearing on the official story of the Evangelion series.
One popular 'asuka y shinji fanfic' could be those that explore their complex relationship in a post - apocalyptic setting. It adds an extra layer of tension and forces them to rely on each other more.
Fanfiction.net is also a good place. There are many different types of 'asuka y shinji fanfic' there. Just be aware that the quality may vary, so you might need to do some sorting through the search results.