Both are authentic. The second hadeeth states that it is most likely in the last 7 nights but doesn’t limit it to these nights as it can be in the other three as well.

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This is permissible because there is nothing in shareeah to indicate the opposite and this is not a form of worship. Ramadan kareem is not correct, rather it should be said ‘Ramadan Mubarak

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It is narrated that the companions may Allah be pleased with them used to congratulate one another by saying: Taqabal Allahu minna wa minkum(May Allah accept this worship from us and you)

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