Could there ever be any logical basis for the thought: "I am untrustworthy"?

I assume your worry is not about whether you are untrustworthy in some areas, or in some sorts of enterprises. As Peter Lipton says, if you are dishonest as a general rule, then plainly you can know that about yourself. And all of us have excellent reason to think that we are not good at many things. But if your question is whether you could have any good grounds for thinking that you are untrustworthy in some very general way (for example, epistemically--in the way you generate beliefs about the world), then I think the answer is also yes. Most skeptical arguments seem to me to give at least some reason for thinking that we are epistemically untrustworthy. Most philosophers these days are not won over by skeptical concerns, but that is not to say that they regard such concerns as logically impossible or incoherent. I think, moreover, that the field of moral psychology gives us some reasons for thinking that we may be somewhat untrustworthy in other areas, as well.

While reading Nichomachean Ethics and Politics , I found myself agreeing with Aristotle far more than I did with Plato when I read The Republic . Can you convince me otherwise? How would Plato have critiqued Aristotle's works?

I would first encourage you to see what is common to what the two philosophers say. Each one thinks that eudaimonia (happiness or flourishing) is what makes a human life good, and each one thinks that the best way to win that goal is to be virtuous. Each also thinks that being virtuous requires acting in accordance with reason. But there are differences, and there is nothing wrong in responding to these with a preference for one account in favor of the other!