Am I entitled to any compensation from the family trust?

No one could really answer this without knowing the specific terms of the trust - what's allowable, for whom it's allowable, etc.

In addition, is it a living trust and if so how has it been funded? Who are the makers and the beneficiaries? Who's the Trustee? All these issues factor into whether or not you can receive funds from it.

Well, certainly. No one can work for free. You need to set up a contract, now, and they need to pay you. It would cost them a lot more to go to a nursing home.