I wish to return to your opening question, Walt, "Time, according to Kant, is a product of the mind. So, without time , without the mind, could the Universe exist at all?"
I find Kant difficult to understand. I have made some good attempts by reading carefully, some of what he has written, thinking about the things he has written such as the question you posed, and by talking to professors who probably have a decent understanding of him. Nonetheless, I am not sure I ever fully and correctly understand him. With this qualification in mind, I will then begin.
It seems to me that Kant's assertion that time is a product of mind, is a necessary consequence of his belief in the existence of a priori synthetic ideas. Simply, he has a belief that there are ideas in the mind, which have no relationship to sense data obtained from the outside world, and which are a product of the mind. He believes, that a person is, or can be, aware/conscous of these a priori synthetic propositions.
In the late A.D. 1800s and early A.D. 1900s, a branch of philosophy was initiated called phenomenology. It states that all acts of knowing are intentional. By this is meant that when we are conscious or aware, we are always conscious or aware of something outside ourselves. A corollary of this belief, would then seem to be that if there was not a something-outside-of-mind which was the source and focus of our awareness/consciousness, we would then have no awareness/consciousness at all; much like the experience, I would think that one has upon awaking from hours of surgical sedation in which the hours seem to just be missing.
Kant's idea is, in this regard, similar to what has been characterized as Descartes' ghost-in-the-machine which he used to describe the human res cogitans (mind) existing within a res extensa (body). Kant and Descartes appear to think that one could be aware of images which exist only in the mind; which have no corollary source or focus on things outside the mind, presented to the mind by the means of sense organs.
One of my professors at the School of Philosophy in Washington, D.C., who after earning his Ph.D. in phenomenology and then serving a stint at the Los Alamos weapons lab; has in his writings described this ideas-as-images-in-the-mind-alone as the "egocentric predicament". He (my professor) asserts that if such a thing were possible (which, as a phenomenologist, he denies); that if such a thing were possible, we would then be in an egocentric predicament in that we would not be able to determine if our acts of awareness and consciousness were of things outside of us nor would we be able to determine the consistency between the images within the mind and the objects outside of the mind to which them might seem associated.
Now, no one I know, actually doubts the connection between ideas we have and the objects outside the mind to which those ideas refer. But, because of Kant and Descartes (et alii) we have become unable and unwilling to assert this connection. The necessary result of this inability and unwillingess, is "When we try to think about human consciousness, [if] we start with the premise that we are entirely “inside,”...we are greatly perplexed as to how we could ever get “outside.” If we are bereft of intentionality, if we do not have a world in common, then we do not enter into a life of reason, evidence, and truth. Each of us turns to his own private world, and in the practical order we do our own thing: the truth does not make any demands on us. Again, we know this relativism cannot be the final story. We do argue with one another about what ought to be done and about what the facts are, but philosophically and culturally we find it difficult to ratify our naive acceptance of a common world and our ability to discover and communicate what it is. The denial of intentionality has as its correlate the denial of the mind’s orientation toward truth."
And of course, the necessary consequence of a "denial of the mind's orientation toward truth" is a community's lack of ability to engage in meaningful political, social, or relational dialogue.
Phenomenology is a continuation of the moderate realism of Aristotle that consciousness and awareness are always associated in some with with things outside of the mind with which we are in contact by means of images/data/percepts funneled through the material senses of the body to the brain and thereby to the mind (the brain in cooperation with the soul).