Do people with alzheimer's disease also get depressed?

Yes. Depression often precedes the development of dementia in alzheimers disease. In early alzheimers it can be difficult to diagnose the dementia though the depression is easier to see. In later alzheimers, the symptoms of depression may be there but a depressive illness is much less common. Some patients with advanced alzheimers benefit from antidepressants though these medications do have risks.
Yes. People with a diagnosed dementia disorder (such as alzheimer's) can and many times will develop some depressive symptoms but they can be very difficult to clarify in the midst of the dementia. Also, many times the initial presentation of suspected dementia may actually be depression in an older individual and must be ruled out prior to diagnosing a true dementia disorder.
Yes. It is often hard to treat and identify, since both can co-exist, however a helpful tip is sudden deterioration in a stable patient with alzheimers.