You could also replace “Can I” with “Do I get to”, which might be less confusing ;)
In a context like this, could "kan" be used in place or "får"? Or would that change the nuance of the sentence?
I think we have pretty much the same situation with these verbs as in English – there's this example often quoted (when discussing the English verbs), where the pupil asks Can I go to the bathroom? and the teacher answers Well, can you? – the teacher would like the pupil to say May I…? but in practice, that is not how the language is used, people do say can to mean may. Same in Swedish, in theory kan should mean 'am able to', but in practice it's used to mean the same as får too.
The verb question is either ifrågasätta (if you mean as in 'put into question') or förhöra if you mean it as in 'interrogate'.
It's analog to the german dürfen. Like "one is not allowed to smoke hier" = "hier darf man nicht rauchen" = "här får man inte röka". Right?