"We are both students."

"somos ambos" or "ambos somos" are both correct however

You only use "ambos" at the end of a sentence when you want to emphasize that both are participating, but you have to add a comma: "somos estudiantes, ambos", otherwise, you're safe saying "ambos" at the beginning of the sentence.

Same question from me: what's wrong with "somos ambos estudiantes" (maybe another person asking will help - there are already 3 others).

De esa forma no me parece, si pones "los" primero sí ó sea "Los dos somos estudiantes."

Yes. "Nosotros" is just the pronoun for "we" or "us," so does not mean "we are." You need a verb with it -- "nosotros somos" means the same as "somos," and the "nosotros" is redundant.

"Nosotros somos estudiantes" is just "we are students", the "both" adds other words (ambos, los dos) :)

You could say "nosotros dos somos estudiantes"

i put it so, too - and it marked wrong. why?

Nosotros doesn't tell that there are specifically two of you. It only stays that there are more than one and you are part of the group. You need a word like ambos or dos to express the thought completely.

"Ambos" already tells you it's more than one person, "somos" already tells you it is "nosotros" so adding "nosotros" is completely redundant and unnecessary and it doesn't sound the least bit natural.

I think it's because it's similar to your job/occupation, and jobs in spanish always use 'ser' not 'estar'

It is a temporary state but it's not an adjective. Only adjectives that describe a temporary state take estar.

Nos means us, not we. I don't know if there are any other problems with your sentence.