Welcome to the board Caring... and yes you are caring. Just the fact that you are here says a lot. Your sister is only in for more and more frustration trying to care of your Mom alone 24/7. If she is yelling at your Mom for extended periods of time... that is abuse.
HN are not the horror stories of the past. Not only was my grandmother in one but I worked in one myself. In the 7 years I was there we NEVER had a bedsore except the ones that came in the facility with the patient. We did heal many. Each patient was fed to their ability to eat and process food. Your sister's frustration level is high and she is exhausted. That can not compare to three shifts of professional staff with a large support staff to help them. A good NH is a not a curse but a blessing.
The community center is right. There are meds that your Mom can take that will calm her down and not make a zombie out of her. Your sister might benefit from some antianxiety meds at the moment as well. She could definitely benefit from some time off.
If your Mom is refusing to eat it may be that she is not feeling hunger or thirst or it might be that she has forgotten how to chew or swallow. Making meal time a battle is only making it worse. Your sister's anxiety will color how your Mom reacts. Giving her finger foods or chunks of foods that she can handle by herself is an option. If she can't chew or swallow them appropriately she should be getting thickened liquids or puree's supplemented with a nutritional supplement like ensure. At some point she will forget how to process food. Force feeding her will do no good because her body can't use what it received. Towards the end, when she has forgotten how to process the food she eats and even before she will drop weight dramatically. There is nothing your sister can do about that. At that point you risk aspirative pheumonia from inhailing food particles into the lungs when force fed.
I would suggest you go with your Mom and Sister to the next doctor's appointment and bring up your concerns. Let the doctor explain this to her with you there. Ask for a written report. Then your sister will have more difficulty negating what he says.
If nothing else works and she continues to take her frustrations out on your Mom you can call your state Social Services for an investigation. That is a dramatic last resort and it may be the end of your frienship with your sister but it will get your Mom the care she will need.
As for yourself..... get the help you need as well. Talk to a doctor and if you don't want medication at least find yourself a good support group. I have a close friend that lost her husband. Even thought she resisted at first, she loves her support group now. They have done for her what this board has done for so many of us. You are working, caring for two kids, and worried about your Mom and Sister. You have a lot on your plate. Give yourself a little credit

You are doing amazing and doing good...
Love, deb