It's not natural-sounding English without the "a". We always use the indefinite article before job descriptions.
(You could make an argument for "She is Mother again," where Mother is being used as her name. For example, "She was not herself last week because of the illness, but now she is Mother again." But that is a little far-fetched.)
It sounds a bit odd, like something a foreigner would say. :)
As a native speaker, I unfortunately don't know English grammar rules enough to articulate why. Maybe it's that "again" tends to emphasize the word that comes before it? With "She is again" or "She again is", it makes me consider a different possible interpretation: "Let me remind you once again that she is a mother."
I saw some people in the comments on another question saying that erneut is more of a formal word meaning something similar to "anew". It could also mean "in a new light" - so the sentence "ich sehe dich erneut" Would mean "I see you as a different person" /"I notice the changes within you"
Native english speaker here. I read "She is a mother again" as a euphemistic way of saying she was pregnant and has given birth. It doesn't mean she stopped being a mother at any point. If you put it in the future tense ("she will be a mother again"), it would be similiar to saying "she is with child." Both of those expressions mean she is pregnant.
Maybe it means "she lost her child, but now she has given birth again or adopted, and so she is a mother again"? I am not a native English speaker, but the context I suggested is the only one I feel fits this sentence. I don't think a woman ceases to be a mother unless all her children die, so if she gives birth to a second child, she doesn't become a mother again since she already is one.
Erneut is both an adjective and an adverb. As the title of this exercise says, it is used as an adverb here. This means you cannot inflect it according to the noun and you have to leave it as it is. If you used it as an adjective (with meaning of 'new' or 'repeated') you would need to change the ending.