Yes. Depending on the severity of the depression, a depressed person might have problems with even basic things as brushing teeth and showering regularly and being able to do household chores.
There are non-depressed people who also struggle with this, I think.
Things like ADHD can make it a real struggle, although depression is a frequent comorbidity.
Yes, that's how symptoms generally work. Symptoms are signs that may or may not point toward a diagnosis. A diagnosis such as depression is something separate that is determined from the collection of symptoms on an individual by a doctor.