1. After Second son's death, who has rights on his part? whether the children of first son or second son's wife?
Upon the intestate death of this second son, his surviving class I legal heirs will be entitled to acquire the property that devolves on his name or as his share from the inherited properties. Here since he has no children, his wife being the lone surviving class I legal heir, his share of property will go to her as his legal heir. The children of first son cannot stake any claim in such properties when the wife is alive.
2. Whether childless widow can ask her rights behalf of husband through court?
She can very well issue notice for partition and separate possession of her share in the properties, if not acceded to, she can approach court of law for relief and remedy.
3. If second son adopt a child after his father's death, he can transfer his part to adopted child?
If a son was adopted properly as per law, the adopted son can have a share in the intestate property along with the wife of the deceased out of the share of properties of the deceased son. The adopted son will lose his rights over the properties from his original father.