No, but in the midst of "unremitting hostility" between their respective governments, a Klingon individual or crew and a Federation one could be forced by circumstance to overcome their differences and accomplish something together.
If we never saw the Borg again, it would be too soon, as far as I'm concerned. Fighting the Borg is not something new and different and exciting that we are missing out on here. It has been done to death.
Again, some of them certainly could. There could be a faction or splinter group that wants to do just that, creating a dilemma for the Federation as to whether to let them go or try to prevent it. Or we could have the reverse of TNG's "The Defector," with a single individual attempting to divulge Federation military secrets for reasons that make it seem to him or her "the logical thing to do." (Kind of like a Vulcan Edward Snowden!)
Yes, and moreover we already know that they did exactly this in roughly the period in question! In "The Cage," Pike speaks of "the time barrier" being recently broken, which would allow the S.S. Columbia's crew to return home faster than they would believe. And in "Where No Man Has Gone Before," the Enterprise is on a mission attempting to probe out of the galaxy for the first time. (Or rather the first time known to Kirk, which immediately turns out to be not in fact the actual first.)
I can never get my head around this notion so many seem to have that if something wasn't mentioned or seen later on this means it wasn't around, or that if it wasn't that this is somehow inherently inconsistent and requiring "explanation." How many species, planets, people, events from TOS were mentioned on later shows? Only a few; many not ever again. Space is VAST. Everything we've seen in all the series and films combined comprises only a tiny slice of it. And even within the specific areas that we have traveled repeatedly, what we've seen in the course of it is in many cases only a tiny slice of what could be going on there. In a way, it often makes more sense for lots of things to be introduced and not heard of again than to have everything be closely connected. (Of course, I do enjoy the dramatic possibilities of having recurring elements and continuing arcs, and I'm sure this show will not be without them.)
I agree with you, yet would also add that even telling stories from points of view that seem to be less central to the "main action"—or in fact are very much central to it in their own small but important way, and have been marginalized by omission because they aren't those of the sort of characters who would be traditionally portrayed as the "heroes" who garner all the fame and glory—can effectively "reshape" the history and universe in the eyes of the audience to a great extent, and that's another part of why they are worth exploring. I think that may be a significant part of what Fuller aims to do in DSC, with the lead character being a 23rd century female Lt. Commander, LGBTQIA and "alien" characters receiving greater representation, etc. Reexamining an era previously depicted through the literal and figurative lens of 1960s society through that of today's has the potential to open all kinds of new windows on it, and to provide a compelling new perspective.