Okay, well, let's assume he dies without leaving a will. KY does NOT recognize common law marriages. Therefore, he has no spouse. Anything he owns jointly with your grandmother becomes hers (say a joint bank account or house). The rest belongs to the children - and their status is a bit complicated if they were born out of wedlock.
Essentially, by not being married, she is not going to get his social security benefits, or have any rights to his property. The problem is, there is a length of marriage requirement for spousal benefits, and off the top of my head, I know there are exceptions, but I'm fairly sure that if they married today and he died tomorrow, she'd still be ineligible. That's one you'd have to go ask someone at Social Security, because it's a complex question, especially since she's apparently getting SSI.
The next problem is that if the parents were not married, then the status of the children under KY needs to be clarified in order for them to be the heirs. Let's assume they are determined to be the heirs, in that case, each child is an heir to the estate in equal portion - if one of them has died, their children get that portion. Now, should the paternity of the children be in disputed under KY law, then grandpa's siblings are the heirs. I doubt that would be an issue unless you have a great uncle or aunt who wants to be a complete ass.
His mining pension is another question. He may or may not have a survivor option there that he put her down for. Also, he may or may not have life insurance, which goes to the beneficiary outside of probate.
Now, grandpa should have Medicare. He may have a medicare gap policy, or Medicaid. He may qualify for Medicaid and not have it. Hard to say. But Medicare pays for a substantial amount of the hospital bills, so while big, it's not likely to be crushing amounts. Should he choose not to continue treatment, it sounds like he qualifies for hospice, which Medicare will pretty much pay for in full.
In terms of the funeral, a direct cremation without any services is the cheapest option, usually followed by direct burial. Go around to the local funeral homes and ask for a price list. In my area, a direct cremation is around $700.
On the UP side, not being married means she has absolutely no connection to his debts, unless they are specifically joint debts. So they won't come after her for the hospital bills. Legally she's his roommate, not family. The hospital WILL go after the estate.
tl;dr - not being married is a huge can of worms for your grandparents. You are going to need to ask questions. Lots of them. Possibly to an attorney. And in the end, it may be best for a simple will leaving everything to her, because it's somewhat likely that she is never going to see any social security benefits from him. And while a full blown funeral can cost thousands, a direct cremation can be relatively inexpensive.