How does one supplicate in prostration? And is it permissible to praise Allah therein and send blessings upon the Prophet, peace and blessings be upon him? And is it permissible to raise one's hands in supplication after concluding the prayer with the Taslim?

The closest a servant can be to his Lord is when he is prostrating (sajid). He may supplicate (du'a) with whatever he wishes, especially for what he is in dire need of. It is preferable to supplicate with the reported (ma'thur) supplications. It has not been reported that the Prophet, peace and blessings be upon him, used to begin his supplication in prostration by praising Allah, extolling Him, and sending blessings upon the Prophet, peace and blessings be upon him. Raising the hands for supplication is legislated unless there is evidence for an exception, so there is no objection to raising the hands for supplication after the salutations (taslim).