Shayne, I'm so very sorry to read all this. My thoughts are with you and your parents. Like Deb, I'll let the people familiar with hospice answer your questions about it.
Just to play the optimist for a moment... there are a number of people here who have had the kind of surgery recommended for your mother. I know it's tough, and with lymph node involvement she would need a neck dissection as well. Is this something you've absolutely dismissed? Yes, afterward radiation and likely chemo are also in the cards, although you say your mother could not tolerate it. I'm going to figure a cardiologist has ruled that out for her (?)
At any event, we're all here to support you in every way we can. It's a tough burden being placed on you. Be strong for them and we'll be strong for you.