Is it obligatory to make up prayers for one who performed invalid prayers due to the emission of wady or madhy, and then made them up with a dual intention, or made them up without a prior intention during ablution?

The believer is obligated to perform the prayer at its prescribed time. If its invalidity becomes apparent, it must be made up immediately before the time expires. It is not permissible to delay it and combine it with another prayer unless there is a legitimate excuse that permits combining them.

When making up prayers, an intention is required for each prayer. It is not valid to combine the intention for two prayers. It is not necessary for the intention to specify "performance" or "making up," so if one intends the prayer, it is valid even if it becomes clear that its time has elapsed.

As for ablution (Wudu'), an intention is not required during the ablution itself. Rather, it is sufficient for the intention to coincide with the opening takbir (Allahu Akbar) or precede it by a short time. One must be cautious of waswasah (whispers/obsessive thoughts) regarding intention, as intention follows knowledge. Whoever knows what they intend to do has indeed intended it.