Thanks Casiopea.
Regarding the desk sentences, we are agreed.

Regarding the first batch, can't a child ask his father that they leave (a party for instance)?
A-I asked my dad that we (he and me and may-be my sister too) leave.
B-I asked my dad for us to leave.
If these two are acceptable, then I wonder whether:
D-"I asked that we leave."
and:
E-"I asked for us to leave."
could not be acceptable.

I think that in all these sentences one could use "asked if" too, but would the meaning be the same? Imagine a child who says to his father: "Dad, let's go, please." How can you put that into indirect speech? Would you use, "He asked his dad if they could leave." in this context?

"If" gives the impression to me that the person is asking for permission, but "that" implies to me that he is suggesting that they leave, but he is in a sort of inferior position.

I might be completely wrong of-course, in which case don't hesitate to tell me.