Upon the death of a loved one, what is the proper course of action needed by a live in caregiver re: home and home equity loan?

As long as payments continue, the lender will not care who is making them. As for how the estate is taken care of, there will be a PA inheritance tax due upon your mother-in-law's passing, so there will have to be an appraisal of the value at that time for the purpose of paying those taxes (4 1/2 percent tax on the net value of the estate, valued at the date of death). For ease of administration, your mother-in-law should find a local trusts and estates attorney and write a simple Will distributing her assets and appointing an executor. Good luck.