It's easy for example to show that if are the minimal polynomials for , , and respectively, then . The conclusion pretty much follows then from the fact that a linear operator where is a -space is diagonalizable if and only if splits over into distinct linear factors.
You can use the same theorem for part two. Namely, since the characteristic polynomial splits into linear factors and its minimal polynomial divides it must split into linear factors.
What do you think for the last part?