Would an anthropomorphic robot who could communicate with and do the bidding of humans qualify Halachicaly as a Golem? Would new rulings have to be devised for their status?
We actually don't have a huge ton of halachic literature about the status of a Golem (though there is the responsum about counting it for a minyan, as Gershon referenced), so questions about robots will probably be handled one at a time. But read on:
Talmud Sanhedrin 65b
רבא ברא גברא שדריה לקמיה דר' זירא הוה קא משתעי בהדיה ולא הוה קא מהדר ליה אמר ליה מן חבריא את הדר לעפריך
Rava made a man and sent it to Rabbi Zeira; [Rabbi Zeira] greeted it but it would not respond. "You are a product of sorcerers!", said [Rabbi Zeira]; "return to your dust!"
As the late Rabbi Dr Azriel Rosenfeld observed, this artificial humanlike thing didn't pass the Turing Test. When/if we develop machines that can pass it, they would be different in that regards from a Golem.
Much, much more of this is discussed in Loike & Tendler, "Ma Adam Va-Teda-Ehu: Halakhic Criteria for Defining Human Beings", Tradition 37:2 (2003). Rabbi Yaakov Emden (shut Yaavetz II:82) says a Golem lacks the intelligence (see article; is intelligence defined by speech, morality, or both) to be counted for a minyan. R' Gershom Hanokh Leiner, the Radzyner Rebbe, infers (Sidrei Taharos Ohalos 5a) that if a Golem passed some intelligence threshold, it would be considered human; if made by a Jew, it would be treated as a Jew and count for a minyan. (Now a Jew "makes" a Golem out of dust much more than he would "make" a robot out of parts, so that last part would be unclear.)
But that should get you started.
So long as the robot is not Mechuyav in Mitzvos you would be unable to use him for a Minyan.