問題 and 疑問: An issue and a remaining doubt
Compare a difficult problem and a no-problem judgment with a remaining question or doubt that leads to recalculation. Separate a doubt from proof that an error exists.
A problem and a limited no-problem judgment
問題 can name a problem or issue to address. The examples include a difficult problem to revisit later and a judgment that a particular situation is not a problem.
「電車が遅れても、時間にはまだ余裕があるから問題ない。」 (Even if the train is late, we still have plenty of time, so it is not a problem.) The no-problem judgment depends on still having enough time despite a possible train delay. It is not an assessment that no problem exists in any situation.
A doubt about an explanation or result
疑問 names a doubt or question that remains. A question after an explanation and doubt about a result concern what is still uncertain, rather than automatically proving an error.
「結果に疑問を感じ、計算をやり直した。」 (I doubted the result and recalculated it.) Doubt led to recalculation. The sentence does not report discovering an error, correcting one, or obtaining a confirmed final answer.
Read the issue's state and the doubt separately
The difficult problem will be considered again later; it has not been reported solved. Conversely, 問題ない does not state that every doubt about an explanation has disappeared.
Identify the issue or inconvenience and the point still in question separately. Even when a checking action is reported, keep a finding about actual error within what the sentence states.
How does having enough time and saying 問題ない differ from feeling 疑問 about a result and recalculating?
Show the answer and why
The first judges that a possible train delay is not a problem because there is enough time. The second reports doubt and the action of recalculating. It does not report discovering or correcting an error, so the doubt itself must not become a finding of error.