1. Zero function is a constant also. No need for the limit definition.
2. Good question! It is impossible for the function to be differentiable on the end points because in order to be differentiable it should continuous first, and in order to be continuous it should be continuous from right and left. But the function is not defined at the right of b and the left of a. This means it is not continuous on b from right (or a from left), hence it is not differentiable.
3. Because this test is given as an application of the derivatives so it should be written so that the student will differentiate in order to test his understanding of differentiation.
and you are right. f is increasing (decreasing) if and only if f' > 0 (f' < 0).