Your question is complex, so I'm going to break it into parts. You ask if the nursing home can force a patient (your mom) to take medication.
described where your mom is as an "adult family home". I'm not clear if these means a licensed, skilled nursing facility or some other kind of facility. If it is a licensed, skilled nursing facility or "SNF"it is regulated by Medicare and Medicaid and they must follow the rules. Those rules do not include forcing anyone to take anything he/she does not want to take. In fact, the rules specify that residents have the right to self determination. They have a choice about what they will take. She can refuse Coumadin. It's probably hard for her when they keep trying to give it to her, and maybe she doesn't know which pill is which.
You need to consult with another MD. This is not fair to your mother. If the doctor is afraid to do as his patient asks, then work on finding a more respectful MD.
Next, you mention a Do Not Resuscitate order. It means that your mother does not want certain things done to her if she experiences heart failure or respiratory failure. She has the right to choose. If she believes that she wants to "give up" she can decide to do that. But, it is very sad to hear that she is so depressed.
If I were her family member, I would want to get a competent mental health evaluation and work toward properly medicating the depression. If she is already taking anti-depressant medication, it may be in an insufficient dose, or the wrong one to give her the desired result: to feel better. Here is where advocacy comes in. You can speak up for her, request the evaluation, insist, and pester whomever you need to to get it done. No one needs to get to this part of life feeling horribly depressed. Please do what you can to use the power of medications to relieve some of that problem in every way you possibly can.
Next, let's look at the problem of her throat being dry, nose dripping and face drooping. Have you gotten an explanation for what is causing these problems? Can they be addressed? It sounds completely miserable. She needs to be helped to drink water every hour and given ice chips for her dry throat. Insist on this. If she is getting help with meals and snacks, good, but if not, be a squeaky wheel.
When I worked in nursing homes, too often the ones who had trouble eating just has a tray shoved at them. When they took a long time to eat or the aides were busy,the tray was just taken away later, unfinished. Step in. Be there a lot. Stand by her side and help her in every way you can. If you can afford to hire a private helper to relieve you, please do this.
Consider hospice care. If you don't know what that is, please learn more. The idea is to keep the person comfortable, not treat their many illnesses and conditions. Perhaps your mother is eligible. If so, Medicare covers it. Please ask for it. Perhaps it is time for her to receive this kind of help in this part of her life.
You mention that you are the POA. I take it that this means you're the agent on her healthcare directive. If so, you've got legal authority to hire a different doctor, demand a mental health evaluation, insist on your mother's rights being honored, and to have your mom's wishes respected so the Coumadin is stopped. An agent on a health care directive has the legal right to say "no" for your mother and they are required to comply with what you say. If it takes a lawyer to enforce mom's rights, see an elder law attorney.
You could move her to a different facility if necessary or if that would make things better. I suggest that you consider these things. She needs your help desperately. It is a heartbreaking situation, and frankly, it makes me mad. I hope you will be strong in your advocacy for mom. I wish you courage and the support you need to keep helping until the end of mom's life. When you look back, you will never be sorry that you kept trying your absolute best. You and your mom are in my thoughts and prayers. Bless you for reaching out.