I do not see a way around you not going to court. I am also concerned as to why your parents would give up custody, and if on active duty you would be granted custody. I suggest you contact the local child protection department where your half-brother lives, but know that this could lead to him being removed from your parents and placed in foster home until things can be worked out.
If your step-father received money from Social Security for time after his death he was not entitled to it and it needed to be sent back. Your mother would need to go into her local Social Security office to see if you qualifies for a percentage of your step-father's benefits---this depends on many factors. If your mother was not named on the bank account, she cannot access it unless she is the Executrix of his will or Administrator of his estate---bother determined by the Probate Court.
You should contact an experienced Elder Law Attorney in your area to discuss the pros and cons of Guardianships. Being a friend and not a family member brings up issues that need to be addressed by an Attorney.
The only way anyone can receive SSI (Supplemental Security Income) is if they are a U.S. Citizen or if they were granted some form of political asylum in the U.S. So based on your question I would say no she cannot receive benefits.
Your child cannot receive the money that her father was drawing on his mother's social security (I guess he was disabled before age 22?). If her father had put in to Social Security on his own, from his own work or if he was receiving Social Security Disability on his own work record then she would probably be entitled to some money depending on her age (either survivor benefits or a continuation of her benefit based on her father's Social Security Disability).
Your $1200 will be considered income by NY. If New York considers a person who receives $1200 a month entitled to public assistance then you can qualify. Getting Social Security Disability is not an automatic bar to getting public assistance if you qualify income wise.