Does alzheimer's have distinct stages?

Sort of. There are a number of different staging systems. If you don't have a specific purpose (like fitting into a medication trial or looking for hospice) I find it useful to look at 3 stages. Mild dementia requires no outside assistance. People with moderate disease benefit from having others regularly check on them and assist with complex tasks. Those with severe disease require 24 hour supervision.
Yes. Alzheimer disease usually starts very mild with short term memory problems. As time goes by over the years additional cognitive problems occur such as: missed bills, car accidents, conversation problems etc. In severe cases patients can not communicate anymore and need help with all their activities of daily living.
There's a New One. Besides the expected mild, moderate, & severe, there is now officially a 4th category --- mild cognitive impairment (mci). This is a stage before mild & the thinking is that here would lie your best bet at preventative strategies (if & when we find some of those).