1. does my aunt and her children inherit any property after the marriage to second party (from first marriage)
So long your aunt remains a widow she is entitled to a share in her deceased husband's property but once she has remarried, she loses the right on the first husband's property.
2. do we need to give anything in writing to family of first marriage on secession of rights of widow and uncles children?
The children will be entitled to the intestate properties of their deceased biological father even if their mother marries another man after the death of their father but they should not be adopted legally by the stepfather.
3. is mandatory to get an agreement for children from second party (widower) after marriage so as to protect rights of children
The rights of the children are already protected by law hence the step father or his family members cannot grab away their properties or do any harm to their interests.