"I will finish my work this week" natural way to phrase it. "I will finish my work OVER the weekend" over is always used for the weekend. " You should hand your work in at the end of the week" natural way to say it "you should hand your work in before the weekend" before Saturday or Friday closing time. "you should hand your work in at/over the weekend'' either is good

I don't think you can say "over the week". In this case, "I'll finish my work during the week" or "I'll finish my work this week" sound like possible ways to say it. Similarly with the second sentence. I think it sounds correct to say "You should hand the work in to the teacher during the week" or "You should hand the work in to the teacher this week". In both cases, the second sentence makes more sense if the work has been ongoing for a while (for example, you've been working on it all term, and should be finished by the end of this week) while the first sentence makes sense if you were set the work on Monday and will work on it and hand it in by Friday.