Is the funeral prayer in which the Imam hastened considered valid? Is it permissible to repeat the funeral prayer for the deceased after burial, over the grave? What are the consequences for someone who does that if their action is not valid?

The prayer is valid if the Imam performs what is sufficient in it. Reciting Al-Fatiha is recommended, not obligatory. Any form of salutation upon the Prophet is sufficient. Supplication is a pillar, and any utterance that qualifies as a supplication is sufficient. Re-performing the prayer is not legislated for one who has already performed it. However, if one re-performs it believing the first one was invalid, there is no sin upon him.