In the state of New York , If there's no agreement in place, is there an obligation for the father of the child to pay child-support during the time the child is attending college.

Yes so long as the child is in full attendance, the father has to pay child support up until age 21 or till said child finishes college, not too sure, but I know the minimum age in school, even a trade school is 21. Hope i was able to help.

If the father has SSI then the child gets SSI also. You have to sign them up at the social security building.

Yup