On knowledge: is there any circumstance in which I am without doubt free from the sceptical possibility that all things are in my imagination only?

There is yet another way to doubt what Descartes supposed we could know with certainty. It seems that one might also wonder if we might be fallible even in the ways in which we categorize our own experiences. Descartes imagines a very powerful malin genie (an evil demon) who would delight in confusing us about anything and everything. Couldn't such a being even confuse us about how to categorize experiences as thoughts (rather than, say, headaches or tickles). To identify something as a thought (and not a headache or a tickle), we have to be able to rely on the conceptual apparatus by which we make such distinctions. It is not obvious that our conceptual apparatus on this or any other subject is invulnerable to error or confusion--especially given the best efforts of some evil divinity!

If you don't have any reasons whatsoever to believe that a certain thing exists, should you deny that it exists, or simply withhold judgment on the question?

I agree with Peter, but would mention a famous debate on just this subject--the debate between William James (in his famous essay, "The Will to Believe"), who contends that there can be non-evidential reasons for certain kinds of belief of the sort you seem to be talking about, and W. K. Clifford in "The Ethics of Belief."

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.