In the first example, I felt that 比べると and 正反対 do not get well together.
(When comparing something, something gradual sounds more natural, say, 父の性格と比べると私はずっとおだやかです/少し頑固です for instnce.)
As Mike pointed out, 性格 is not the only thing that can be "completely opposite" between you and your father.
Once sentence is hardly a context, but if you are talking about 性格, you can omit it. If the sentence is stand alone or the beginning of your talk, you need to say 性格 to make it clear what you are talking about.