Does the alzheimer's gene correlate with any other types of diseases like cancer?

No clearcul relation. Several genes cause early onset, familial alzheimer disease that contrasts with usual multifactorial, older age onset of the disorder. If you enter omim in browser, search on alzheimer, you can see descriptions like presenilin-2 that can cause heart problems as well as alzheimer. These alz-predisposing genes are not directly related to cancer but may accelerate aging and general risks for cancer.
No. This question seems to presuppose that most alzheimer's disease is genetic. Only about 3 to 5% of alzheimer's disease is of the hereditary variety. Most of it is just wear & tear on the brain.