Question 1. In the trust that goes solely to my brother, in the event
either my mother or father are dead, how is that asset split up? Can a
trust dictate how an asset if split up if I am not on the trust.
This is governed by the terms of the trust. The trust can say whatever the people who wrote it wanted.
Question 2. How is the debt handled? The asset still has debt on it,
who is responsible for the debt? I am thinking the answer to this
question depends on the answer the Question 1.
The trust owns the property subject to the mortgage. The people who originally signed the mortgage (presumably both of your parents) are also personally liable to pay the mortgage in the event that the debt is not fully satisfied in a foreclosure sale, or in the event that the mortgage creditor prefers to sue the mortgage debtors individually in the event of a default. After the death of the mortgage debtors, the mortgage company can file claims in their probate estates for repayment of the mortgage related debt if it does so in a timely fashion, although it would be more common to simply foreclose on the real property and obtain repayment of the debt that way.
The death of the mortgage debtors is almost always an event of default under a mortgage, so if both mortgage debtors die, the mortgage creditor can usually force the trust to repay the debt, either by selling the house or by refinancing it.
Question 3. They currently live in New York state, I found an article
that New York state for nursing home care at more that 12k per year.
If they were to move to a cheaper state like Alabama what are the laws
or rules, ie. how long they have lived there, assets.
This question verges on incoherent. But I think I understand what you are trying to ask.
In every U.S. state, the joint federal-state program known at the federal level as Medicaid, although it has different names at the state level (California, for example, calls it "Medi-Cal"), will pay for nursing home care at approved nursing home facilities if the applicants meet state eligibility requirements. This generally involves an income test, an asset test, and a determination that the applicants have not made disqualifying gifts to trusts or to third-parties within the past five years.
The income test is based upon state median or average income and is usually lower in a state like Alabama that has a lower median state income, than it is in New York, that has a higher median state income. For retirees, the main kinds of income considered are defined benefit pension plans payments, annuity payments, and investment income.
The exact details of the asset test vary somewhat from state to state. Generally speaking, New York State is among the most strict in this regard.
The basic outlines of the disqualification for gifts made within five years to others including trusts before applying for nursing home care are similar in broad outline, but the consequences for violating the rules are a function of the average cost of nursing home care in a state, which varies considerably. Generally, a gift of a fixed dollar amount gives rise to more months of disqualification of Medicaid in a state where the costs of nursing home care is low than it does where the costs of nursing home care are high, because the amount of the gift made in the past five years is divided by the average cost of nursing home care at the time the application is made, to determine the number of months that the applicants are disqualified from Medicaid.
More generally, setting up trusts intended to qualify you for Medicaid is a highly technical matter with severe consequences if it is done incorrectly, and I would hope that your parents did so while conferring with a competent elder law lawyer. If they did it themselves, there is a high chance that this will have very severe negative consequences.