Which are correct:
1-I asked my brother that we should help our dad.
2-I asked my brother for us to help our dad.
('we'/'us' could be 'me and my brother' or 'me and some other people'. In other words I said to my brother 'Let's help our dad', or something like 'Let me and our sister help our dad'.)
3-I asked that he should leave.
4-I asked for him to leave.
(Instead of 'I asked him to leave'.)
What about this;
I suggested to my brother that we help our dad.
3-I asked that he should leave.
4-I asked for him to leave.
(Instead of 'I asked him to leave'.)
They both sound wrong to me. If you were wanting to say that another way, you could try;
I demanded (that) he leave.
I suggested (that) he leave.
Or simply, "I asked him to leave."