1. What would be the rights of a daughter in the self acquired property of her maternal grandfather who died intestate ? Mother not alive.
Firstly the grand daughter has no direct claim for a share in the maternal grandfather's property even if the grandfather died intestate. Her mother only has a right to a legitimate share in the said intestate property. As you have stated that the mother is not alive, if she had not availed her share in the intestate property during her life time, her share of property from her her father's estate shall devolve on her own legal heirs i.e., her children and her husband. That way if you are her daughter you are entitled to a proportionate share out of your deceased's mother's share in the property. You can file a partition suit seeking your share and separate possession of the same.
2. My paternal great grand father (GGF) bought property ( self acquired ) and gave it on lease. After his death my grandfather ( GF) his only son,inherited it and renewed the lease. My grandfather died intestate leaving behind two sons ( S1 and S2 ). S2 was my father. The property was not partitioned between my father and his brother. Both of them died young before 2005.
Presently, is that property ancestral self acquired or joint family? Iam a daughter and what is the share Iam eligible for?
Though this can be termed as ancestral property since it has fallen into the third generationas it is, the papers are to be seen because the daughters became entitled to a share in the ancestral as a coparcener with effect from the state amendment of the succession act from the year 1989. Even otherwise you are entitled to a share out of your father's share in the property since he died intestate.
You may decide to file a partition suit claiming your legitimate share.