A lot to take in!
In English "the day after", "the next day" and the "the following day" mean the same. Likewise "the day before" = "the previous day". In French, do le lendemain, le jour d’après and le jour suivant /la veille, le jour d’avant and le jour précédent differ from each other in meaning or mainly in register?
Secondly, from the point of view of today, are l’après-demain and l’avant-hier used in conversation?