Yes.
Dr. Robotnik
Knuckles (Formerly) Shadow Dr. Robotnik
Dr. Claw was the arch-enemy of Inspector Gadget.
No Shadow was creation by Dr. Ivo Robotnik's grandfather and given DNA from Black Doom. They are in no way related.
That depends on the canon: In the Fleetway Sonic, Dr. Ovi Kintobor is who would become Dr. Ivo Robotnik due to his machine going belly up. In the SatAM and Archie Sonics, Robotnik's real name is Julian Kintobor. Also in the Archie Sonic, the Moebeus conterpart of Robotnik is called Kintobor. Note that Kintobor is Robotnik spelled backwords.
Dr. Ivo Julian Robotnik
Dr. Eggman and Dr. Robotnik are the same person, Eggman is his nickname. Gerald Robotnik is Ivo Robotnik's grandfather. Ivo's full name: Dr. Ivo "Eggman" Robotnik Gerald's full name: Dr. Gerald Robotnik
Technically, Shadow is Dr. Robotnik's uncle.
Dr. Robotnik
It all started when Dr. Ovi Kintobor turned into the evil Dr. Ivo Robotnik.
Dr. Robotnik's Mean Bean Machine happened in 1993.
You can refer to him as either. His full title is Dr. Ivo "Eggman" Robotnik. Story line goes, his real name is Dr. Ivo Robotnik, later got his nick name "Eggman" from Sonic. Character creation goes, SEGA of America named him Dr. Ivo Robotnik, and SEGA of Japan named him Dr. Eggman.