Is it possible to write this answer instead the one given in that test
My mother fell ill last week. I am making all the meals instead of her. She still does not feel well so she is seeing a doctor tomorrow . I hope she will get better soon.
Can it be possible to change does not feel by is not feeling well and she will get by will be getting
because both actions are not realised at moment of speaking they are still in progress
Your original statement:
My mother fell ill last week. I am making all the meals instead of her. She still does not feel well so she is seeing a doctor tomorrow. I hope she will get better soon.
Your proposed statement:
My mother fell ill last week. I am making all the meals instead of her. She still is not feeling well so she is seeing a doctor tomorrow. I hope she will be getting better soon.
About your first change, it is evident that you should also swap the "still" and "is," i.e "She is still not feeling well, so..." (That comma will help phrase things better, in my opinion.)
I believe that this replacement is perfectly acceptable, both in terms of grammar and how it would sound to a Native Speaker.
Your second change does seem to be grammatically correct (at least according to Grammarly) but does feel a bit too awkward. I believe it stems from the fact that it is longer than it has to be, and strays away from the typical "get better/well" phrase which is used ("getting" is not used as often, in this context.)
In fact, a valid change to the last sentence could be "I hope she gets better soon."
