What happens when someone files a petition with the probate court for daughter not to be administrator of the estate?

Not necessarily. The Court will make a decision based on a number of factors, including which person appears to be most qualified to carry out the duties of administrator. If you want to maximize the chances of your sister being appointed, you should engage an attorney with experience in probate litigation to retain you. Let me know if I can provide other assistance. Best of luck.

No.

Obviously, you have not found a will. When there is no will, a court looks at a variety of factors including relationship to the deceased, beneficiaries of the estate, capabilities with respect to handling the estate's affairs.

Please be advised that the foregoing discussion is a generalized and hypothetical answer based upon incomplete facts and is not intended to serve as legal advice to you, nor should you regard it as an attorney-client communication or as creating an attorney-client relationship. If you desire legal advice concerning the situation you have described you should contact an attorney who has substantive experience in the fields pertaining to your question.

This answer is not intended to provide you with specific legal advice regarding your situation, or to create an attorney-client relationship.

In Georgia, assuming that there was no Will (as your question implies there was not one), the only persons who would normally have any legal right to object to the appointment of an administrator of the estate would be your father's heirs. If your father was not legally married, then his children would normally be his heirs (assuming he had no predeceased children - in that case, grandchildren by the deceased child would also be heirs, if any). An unmarried, live-in girlfriend would not generally have any rights as an heir, and therefore should not be able to interfere with having your sister appointed as the administrator of the estate, if all of the heirs agree to have your sister appointed.

Georgia abolished common-law marriages prior to 1998, so unless your father and the girlfriend were living together prior to 1998, or they began living together in another state and later moved to Georgia, she likely does not have any potential claim as his spouse. You should consult an attorney with specific experience in this area if you want an exact answer as to what potential rights the girlfriend may have, however, because you will need to let the attorney know all of the relevant facts and circumstances, in addition to the information posted in your question.