‘I doubt if he did it’? Sounds way less natural to me than ‘I doubt he did it’ (rejected) or ‘I doubt that he did it’.
To me all three sound OK, although "I doubt he did it" sounds most natural and "I doubt that he did it" sounds more correct than "if".
All versions mentioned so far work.
"I doubt he did it" set as the best answer.
Yes, but it seems to me that this could change the meaning a bit.
"że to zrobił", especially with specifying "że on to zrobił", can mean "that he is guilty of what they accuse him of doing". Like "I doubt he murdered her, I think he's innocent".
It can also mean I guess simply "I don't think that he did it", with 'it' that can be just 'cleaning his room'.
"czy" only has the second meaning in my opinion.