1. "Neither of the twins" also means "none of them". Yes, it's true, but if the total of a thing is zero (not one or more than one), it must be in the singular form.
2. In this sentence, "were gathered" means that the entire staff were called to gather (被聚集在操场), it's a passive voice usage.
However, if we use "were gathering", it means when it is recorded, the gathering was in progress(聚会在进行中).
The words "were gathering" is often used in such sentences as: The entire staff were gathering when we passed through the field.
3. In the third question, we could not use the word "gathers" because it has mentioned that it happened on yesterday, but not today. Hence, we need to use past tense verbs, while "gathers" is considered as a present tense verb.