1. Succession Certificate is required for claiming movable properties like bank account, shares, FDs, MFs etc. which takes at least 6 months to obtain from the Court by filing an application. Whereas legal heir certificate is required to claim immovable properties like land, flat,house etc. which can be obtained from the local Municipal corporation through ward counselor or Tehshilder with in 7 days.
2. No letter of administration is required here since there is no will executed by your demised father.
3. There should be a settlement deed registered by and between your mother by which you two will settle for the title of individual ownership of the title of your deceased father's property either in equal or on agreed proportion. This will make you the recorded/registered title holder of a portion of the said property.
4. Based on the said registered settlement deed, you and your mother can deal with the said property as per your wish.
5. For selling the said property, the said settlement deed is not required and you and your mother can register the sale deed since you two are the sole legal heirs of your deceased father's properties.