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It does what it says, It's returned to the battlefield (in what state is not relevant) until end of turn and then it gets exiled. I don't think it being an enchantment has an impact on the "exile it at end of turn" part.

All Goryo's Vengeance does is return a card to play with an added stipulation. The Gods are creatures in the graveyard so it's a valid target for the spell.

They will come into play with the additional text of "Haste, exile at the beginning of the next end step" The fact that it's not a creature is irrelevant to both of those added abilities. The haste doesn't do anything but it will still have it so in the event that it's turned into a creature before the end step, it could attack.

The "that creature has haste" is just a shorthand reference to the card that was the target. It doesn't care that it's not actually a creature when it comes into play, it is only referring to the card that was a creature when it was chosen as the target.

A similar situation arises when you animate a non-creature and kill it then imprint it on Mimic Vat. Mimic Vat doesn't care that it has a non-creature card imprinted on it. It was a creature when it died so it can exile it and any tokens that you create will be whatever that non-creature card is. It will still have haste and will still be removed at the end of turn even though it's not a creature.