宿題 and 課題: homework and tasks
Compare homework with the broader category of tasks or issues.
Homework assigned in class
宿題 (しゅくだい) is homework assigned for outside class. 「宿題を家でしました」 says that the speaker did it at home. The word itself does not require doing it only in one's own home.
「宿題を出しました」 can mean submitting homework for a student or assigning it for a teacher. Identify the subject first.
Something to carry out or solve
課題 (かだい) is an assigned task or an issue to address. 「レポートの課題が出ました」 means that a report assignment was given.
「交通の問題が町の課題です」 says that traffic problems are an issue for the town to address. 課題 extends beyond homework to social or work-related challenges.
Completion versus submission
「宿題は終わりましたが、まだ提出していません」 means that the homework is finished but not yet submitted. Finishing and handing it in are separate actions.
「課題に取り組んでいます」 means working on a task. It does not guarantee completion or success, so do not add an outcome.
Does 「課題が残っています」 necessarily mean homework was not submitted?
Show the answer and why
No. It means tasks or issues remain and can concern school, work, or society.