I would just like to ask if the part of the sentence marked bold is correct.
My friend started her chemo yesterday and will be checked next month to know if she will have been responding well to the treatment.
It's grammatically valid as far as I know, but it's not something a native speaker would be likely to use, certainly in that context. It would more likely be "to see if she has been responding". That's because it's acceptable, and more natural, for the tense to be relative to the point at which the checking happens. Also because to see is more natural to use in that context than to know. If you want something more formal, it should be to find out. To know is not generally used in that sense, because it does not connote the gaining of knowledge, only the possession of knowledge.
The future perfect progressive is more used for actual statements of knowledge, or predictions, rather than things that are to be found out. "She will have been running for an hour at that point, so will be tired." In a conditional or other if phrase, it would be "if she has been running for an hour, she will be tired".