Okay, I'll bite. These are not necessarily 100% correct.
What was I supposed to have done?
¿Qué es lo que se supone que (yo) tenía que haber hecho?
When was I supposed to have gone?
¿A qué hora se supone que (yo) tenía que haber ido?
How was I supposed to have known?
¿Cómo se supone que (yo) tenía que haberlo sabido?
It seems to me that the "to have <past participle>" could often be the simple present in English, with essentially the same meaning. In other words, these are almost the same in meaning:
What was I supposed to have done? == What was I supposed to do?
I think the Spanish version could also be simplified along those lines:
What was I supposed to do?
¿Qué es lo que se supone que tenía que hacer?
When was I supposed to go?
¿A qué hora se supone que tenía que ir?
How was I supposed to know?
¿Cómo se supone que tenía que saber?
A few other notes in no particular order.
- "When" can be "a qué hora", "cuándo", or "en qué momento", depending on context.
- There are other ways to say similar things without "suponerse."
- It might also be approriate to change the "tener que" to "deber" or "deber de."