What is the ruling on someone who enters the mosque with the intention of performing a fard (obligatory) prayer, then it becomes clear to him that it is a funeral prayer? Should he change his intention and pray the funeral prayer, then the fard prayer? And should he join the funeral prayer if he has not yet performed the fard prayer?

Transferring intention from one obligatory prayer to another invalidates both obligatory prayers. The first prayer becomes void because its intention was cut off, and the second is not valid because it lacked intention from its beginning. Therefore, what the questioner did by performing the Dhuhr prayer after the funeral prayer was correct. As for the funeral prayer, it is invalid because its intention was not formed at the time of ihram (definition), and its re-performance is not possible. It is better to join the funeral prayer for fear of missing it and for the great reward it contains.