Sure. Especially if this person beats themselves up all the time over a certain situation, and you know how they feel because they've plagued your friendship with this over and over again. At some point, even though you know what they're experiencing (emotional pain) you are allowed to say, I have no more sympathy for you because you keep doing this to yourself (a hypothetical here) or let that other person do it to you again, and you already know how you're going to feel. So I have to ask, why do you keep doing this? Like that.
Yes. One doesn't preclude the other.
You can't, at least not unless you are totally without feelings, or if you take the meaning of empathy in a truly objective way.