Translation:I doubt whether the story is true.
I like how surprised I was to see ĉu used in something other than a question.
If you use mobile, you will see that earlier. I use mobile while I'm away from home. :)
Could you use "ke" in this instance? As in "Mi dubas ke la rakonto estas vera"?
It does not mean the same thing: - "mi dubas ke la rakonto estas vera" = I doubt that the story is true (so I don't think it is). - "mi dubas cxu la rakonto estas vera" = I doubt whether the story is true (I suspect it might not be true but I am really not sure). The two sentences don't carry the same degree of doubt: this first one shows a stronger doubt.
Yes, for reported-speech questions.
In direct questions, it's not translated.
Ĉu li venos? - Will he come?
Mi ne scias ĉu li venos. - I don't know whether he will come.
Is it okay to write the second sentence like this?
Mi ne scias se li venos. - I don't know if he will come.
No - that would mean "If he will come, I don't know", i.e. whether you know or not depends on whether he comes, because in the case that he does come, you will not know.
"Se" really does mean "if".
You could use it in sentences such as "Mi legos libron se pluvos" (I will read a book if it rains (will rain)".