No. Sounds like you may be talking about someone with aphasia, where the part of the brain involved in understanding or generating speech isn't working, perhaps due to a stroke. Alzheimer's patients have trouble with memories, so they know who they are and where they grew up, but can't remember what they had for breakfast.
Possibly. It is hard to know if somebody who cannot speak knows who he/she is. People with alzheimer's disease often consider themselves younger because they live in their past.
Not usually. That is more common in other types of dementia though it is not impossible. The first and primary deficit is problems forming new memories. Word finding problems become more notable over time. However speech is relatively preserved in alzheimer's.
Not exactly. Alzheimers dementia involves impairment in cognitive abilities and memory, and is progressive, ie it gets worse over time. Most people with alzheimer's tend to retain their speaking ability. Whether they can recall their own name is usually a feature of how advanced their illness is, tending to occur in a more advanced state.