1. My first reaction to the image of someone putting the cat in the box: The poor cat! More to the point, though: Yes, you are correct, anyone who has ever observed a cat in the same room with a box knows that the cat climbs into the box all on its own. Similarly, the world is alive: its constituent beings are active, and (I mention this in case it matters) often have their own kinds of volitions, regarding climbing into boxes and the like--they probably do that consciously or with some type of intent or awareness. So, yes, the world is informationally organized into systems of knowable facts; there are meta-organizations known as human ethics; and there does not have to be a sovereign author.
2. I agree that, if God exists and creates the world, then God can create any and all objective facts, not just the moral kind. (Whether or not God does create all facts, he certainly can create any fact he likes.) But this draws my attention to the distinction between facts and knowledge. Under some descriptions of knowledge, our knowledge is a kind of shadowy, insubstantial representation of a fact. So, God may create a solid thing in the real world, but that is not the same as our knowledge of that thing, that is, our own logical reasoning, sense impressions, memories, artistic representations, interpretations, etc. God makes the cat and gives me eyes, ears, hands with which to know the cat, but God may have not dropped "knowledge" of the cat into the box that is my brain. The actual cat is God's (because God is the ultimate author, owner, and arbiter of reality), but my knowledge is mine (because it consists of my stories about reality).
This might lead into the Platonic definition of knowledge as a justified true belief. The theist might argue: My beliefs, clearly, are mine, insofar as they might be fantasies or falsehoods. And these beliefs might, coincidentally and happily, also be true. But the justification that transforms them into full-fledged "knowledge" is the way in which they bump up against and draw from God's reality. In that sense, my full-fledged knowledge is not purely mine, but is co-authored or co-validated with God. In other words, I have God to thank that I have ways in which to truly know something and not merely be accidentally correct about it. On an even more elementary level, I also have God to thank that there exists anything in the first place about which to be correct, regardless of whether I can justify my correct beliefs. What I mean is, if I guess that a bridge will hold my weight and it does hold my weight, I can thank God that there is a hard reality of the situation, and this is preliminary to and apart from a second gratitude to God for my having had good reasons to believe the bridge would hold my weight. (This is what I imagine a theist might say.)
3. Since clearly "non-believers are capable of moral actions and moral reasoning," you suggested this should make us question whether it matters if anyone appeals to God in our moral justifications, and I agree with the direction you are leaning with this question. As an atheist, I'd say the concept of God doesn't add anything to moral process or output. God is an unnecessary add-on. We arrive at moral conclusions and actions without God. Atheist conclusions/actions may be identical to theist ones. Or, the atheists' conclusions/actions may be different, but that does not mean they are worse. They may be better, or perhaps simply different.
4. I agree that the question of the origin of morality is not adequately answered by the response: "God." The way I describe it: The question is generally a "How," as in, "How can moral facts be formed in the world? How can moral ideas enter our heads?" Therefore, to answer with a "Who," essentially saying "God did it," is abandoning the task of answering the real question. Analogy: If I am rolling a little wheelbarrow around a construction site, and it's my first month on the job, and someone asks me, "How is that skyscraper being built?" and I shrug and respond "The construction workers are doing it," that is a true statement but does not answer the question. It's a way of saying "I don't know. I'm only partly engaged in the collaborative work, and I'm not fully aware of the entire scope of the project." Which is true of moral knowledge, too.
