false. for a vector field we must have but in here we have

true. properties of surface integral. see your lecture notes!

2. Let D1 be an oriented surface and D2 be the same surface with opposite orientation. Then, for every integrable function f, we have why?

true. the divergence theorem + see my answer to part 1 of your problem.

3. For any smooth vector field F on a torus is zero

true. by the divergence theorem the value of your surface integral is

4. Let and be the sphere oriented with outward-pointing normal. Then, the flux increases with the radius R.