A "people" is a cohesive group of individuals who self-identify and can be distinguished in point of fact from other groups through measurable traits or factors. Those traits have traditionally included culture, language, religious beliefs, geographic locality, ceremony and ritual, a system of laws, worldview, specific ceremonial practices, system of myths and legends, connection to monuments and antiquities, political views or aspirations, tribal and familial relationships and probably a few things I've missed. A "people" typically has a number of these different traits and can be readily distinguished from others, even if they may share some similar traits.
Who gets to decide whether or not they are a "people"?
Tough question.
In the absence of malice, I'd say that only the people can decide if they are different from all other people.
That said, malice exists and thus an objective standard would be a reasonable starting place.
And that said, recognition is also a factor, in that if there is no way for a reasonable person to distinguish between your people and another people...um....shrug.
At what point do they become a "people"?
When they are sufficiently distinguished, in point of fact, from other people.
Do they need a unique culture
Depends on what they "need" it for. What's the purpose of being a "people"?
(and what "defines" a unique culture"?)
Language. Ceremonial practices. Life event practices. Celebrations. Holidays. Religious practices. System of laws. Special diets. Distinctive clothing. Rituals and ritual objects. Myths and stories. Moral precepts. Probably some I've missed.
Do they need a unique language?
As an objective requirement? No. They very often do, however. Its a definitive marker, imo.
Do they have to have had a nation?
As an objective requirement, in the modern sense? No. This presumes that new cultures and "peoples" can not come into being which is sort of ridiculous. On the other hand, most of today's "peoples" do actually have a some sort of history as a self-governing entity.