My sister lives next door.
My mother went to my sister's house.
My son is looking for his grandmother.
"Where is grandmother?" my son asked.
"She went next door."
"She is at next door."
After the primary auxiliaries "is", do we need to add "at"?
She is next door or She is at next door?
In general when describing a person's location we say they "are in" or "are at" some place. "She is at Aunt Sally's house." "He is in the store." "He is at work." Etc.
If we are using a preposition to describe the place, either as a separate word or part of a compound word, we do NOT use "in" or "at". In this case we don't need two prepositions. "He is downstairs." "She is overseas." "The treasure is under the hill."
"Next door" is something of a special case, an idiom. It does not take "in" or "at". We say, "She is next door." (While "next" is generally classified as an adverb or adjective, arguably here it is being used as a preposition. Idioms can be difficult to classify.)
The correct phrase to use is
She is next door. She is at Auntie's.
About your question on using "at", if someone is knocking on your door, you would say
They are at the door.
