Can the diabetic drug metformin suddenly start causing diarrhea after being on it for a year?

Yes. This is unlikely, but possible if you are taking the same preparation. However, different manufacturers products can have different effects in different people, check with you pharmacist to see if you have been on the same product.
Unlikely. Anything is possible, but this is unlikely and before stopping your metformin you should be evaluated for other possible causes of the diarrhea. If none are found, then changing your medication may be a reasonable option.