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.
'to question someone' is usually a pretty formal or serious event. It's usually used to mean 'interrogate'. Police might question a witness, or a lawyer might question someone during a trial.
You might hear it used in informal settings (eg:- She questioned her daughters boyfriend after dinner), but even then it has connotations of asking a lot of questions to try and ascertain someone's character or motivations, not just a single question about anything at all.