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Hōjō Shigetoki died in 1261.
Hōjō Shigetoki was born in 1198.
No, Hojo is related to Akitoki Hojo, who appears later in the series. Akitoki is an ancestor of Hojo, hence the close resemblance.
Tsukasa Hojo was born in 1959.
Yes and no, Inuyasha kinda meets Hojo in episode 128, Inuyasha and the rest meet Hojo's ancestor in the fuedal era.