I am a teacher and try to be open to issues my students have, but without actual documentation of accommodations, how far I can go with just accepting a student's reasons is fairly limited. The fact is that I have objectives for my class, and in order to be able to evaluate your progress in my course, you do have to do the work within a reasonable time.

As far as how much detail to tell a teacher, you really don't have to provide a great deal of detail. In fact, less is probably better. Things happen. Most of us understand that and we don't need great detail. Privately speaking to the teacher and requesting one extra day is usually reasonable if it doesn't happen often. It is the student who repeatedly asks for that kind of extension that creates a problem. Without accommodations documented by the school, our response after making that exception once or twice will be that we are being taken advantage of. You may not like to hear that, and that may seem unreasonable of us. But teachers do get taken advantage of if they don't set boundaries around their assignments and due dates.

I would strongly recommend you go to your counseling office and discuss getting accommodations on file. Since it sounds like there are times when you may need an extra day for an assignment if a crisis occurs the night before an assignment is due, extended time (usually only one day) seems to be a reasonable accommodation. If concentration during class, particularly during testing situations is a problem due to depression, having extra time on tests (usually time and a half) is fairly common as an accommodation. The fact is, your current teacher may be willing to work with you, but future teachers may not be even mildly accommodating without some documentation that they must do so, and without those accommodations, technically, they don't have to be.