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.