In reply to your first question, you and your sister along with your mother are all legal heirs to your father's property, meaning you all have equal rights and share (ie; 1/3rd each)to the property owned by your father (including the property sold by your mother without informing or taking permission from you two).
To your second question, your mother can give her self acquire property only or her her share of property to anyone she wants. However, that does not include the property which she has purchased by selling off her husband's property illegally.
In reply to your 3rd & 4th question, immediately file a partition suit and also challenge the illegal sale of your father's land claiming share for you both. As an interim relief you can ask the Hon'ble Court to restrict your mother from further selling or transferring of your father's property till the matter gets decided by the court.