No, none of the five Exodia cards are considered god cards. They are not illegal in any form of play, they are Limited.
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Exodia does nothing when summoned to the field, there's no real interaction between him and a god card. He might be able to defeat Ra if its ATK hasn't been boosted, but will probably get crushed by Obelisk.
No. The god cards are stronger than Exodia.
No starter deck or structure deck has every exodia card
Yes, there is no other card more powerful than Exodia.
The card named 'Exodia the Dark Master' is a counterfeit card, and not worth anything.