You have to offer the prostration of forgetfulness before Salam. However, if you did not offer it, the salah is valid.

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Yes, generally you are right. However, if the pillar was performed during the prayer itself, the prayer is still valid as in the case of someone missing a sujood and remembering it before he concluded his prayer and hence he added another whole Rakah in exchange for the invalid Rakah.

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No, but you will be losing a great deal of reward and no house would be built for you in paradise.

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