1) If my mother continues to live with my sister, will my sister's earnings be included in countable income in determining A&A?
Your sister's income does not count in determining your mother's entitlement to A&A. I am assuming your sister is not a dependent of your mother. She is over 18 and living on her own?
2) Is it best for me to deal with the VA on behalf of my mother since I will be completing the paperwork. If so, what VA form needs to be filed to do this?
If you are going to be doing all of the paperwork and any follow-up work on the claim then I recommend you submit the VA form 21-22a.
3) My mother lives in GA. I see from some of the posts that filing in another state might be faster. Can this be done or do you have to file in the state in which the veteran lived?
The claim must be filed in the state where the claimant resides. You can use your mailing address for correspondence from the VA. I recommend do so, to make sure you get all correspondence in a timely manner. Most VA forms and correspondence have to be replied to within 60 days.