Since the din that a bris is Docheh Shabbos is learned from the pasuk וביום השמיני ימול בשר ערלתו, ויקרא יב ג, from which the chaz"l make a drosho "ביום השמיני אפילו בשבת", (שבת קלב ע"א), therefore a bris is only docheh shabbos if it is "Bayom Hashmini". Simply, this means, that Shabbos is the 8th day. If a baby was born on Friday, then the Yom Hashmini, the 8th day, would on the following Friday.

May a Mohel who was Mekabel shabbos early do a bris on a child who turned 8 days old on Friday? or do we only say that a bris may be done on Shabbos only if the "Yom Hashmini" is Shabbos as it would seem from the drosho of the pasuk??

Since the rule that a circumcision pushes off Shabbos is learned from the verse וביום השמיני ימול בשר ערלתו, ויקרא יב ג, from which chaz"l expound "ביום השמיני אפילו בשבת", (שבת קלב ע"א), therefore a circumcision only pushes off shabbos if it is "Bayom Hashmini". Simply, this means, that Shabbos is the 8th day. If a baby was born on Friday, then the Yom Hashmini, the 8th day, would on the following Friday.

May a Mohel who accepted shabbos early do a bris on a child who turned 8 days old on Friday? Does that qualify under the rule of 8th day even on Shabbos (since it is now still actually the 8th day), or perhaps not since the eighth day is Friday and not the day of the week called Saturday/Shabbos even if the Mohel accepted the holiness of the holiday of Shabbos early? Perhaps it would be different if the Mohel is the father who is personally obligated here?