In English people say "I'm doing well" or "she's doing better" but in that context "going" is only used when one would use "it's," "it is," or even sometimes "things are" as in "it's going well" or "things are going well."
However, just because people say going in such a context does not mean that it is correct grammar. People colloquially use incorrect grammar all the time in speech. For example, starting a sentence with "me and my friends" rather than the proper "my friends and I."
On Duo, just try to think about what's grammatically correct, not verbally common. :)
It's a fixed expression so what you think of as "correct" grammar has little to do with the validity of the expression. You're thinking of semantics, and "doing" is not really any more fundamentally logical here than "going" so the only real answer is "do native speakers use this expression?". Look around the comments here from native (presumably mostly non-American) speakers to answer that question.
When you want to make a statement about correct or incorrect English, it is important to say "American English" or otherwise. Remember that America is not the only country to speak English. We are not even the first country to speak English. And remember that this app is teaching French to English, not French to American English, although it seemed favored, I believe.
When someone says my mother is better without any other context, the takeaway meaning is that she is feeling better. If she was in a state where she was getting better without feeling it, the speaker would qualify the phrase since it would be misleading to not include the mother's contrary perception of her condition.
Not in this case. "Va" is in the present tense. You are asking for the future tense.
I am uncertain of how to accurately say this with the usual future tense (that is, using the conjugated "aller," to go, as a helping verb), however, I believe the Imperfect or Imparfait (the other way to do the future tense) form is "allait."
So, "Ma mère allait mieux."
As a native English speaker, however, I am not wholly certain whether this is 100% proper french, as in I don't know if you would still use "aller" for the future form. But, I believe that you would.