Yes, that is right. Sorry I doubted you, but there are people on this forum willing to chop off your left foot in defense of the "scientific fact" that a precise dividing time exists. In fact, you are quite a rarity in my experience.
But it's not clear that 'nature at an instant' has any actual meaning anyway. No physical measurement of which we're aware is capable of recording an interval of time shorter than the Planck time, which is approximately 10-43 seconds, and, of course, we're nowhere near being able to measure so short an interval.
Maybe, but the continuum doesn't require real-number-line kind of smoothness. The fallacy of the beard (when describerd as plucking hairs one by one) is a good example of a continuum of discrete intervals. Our current technology is resolute enough to see that there is no sharp division in the life cycle, because we can see short enough intervals to know that no two adjacent time points at our best resolution are significantly different.
...the continuum doesn't require real-number-line kind of smoothness...
I prefer to reserve the term 'continuum' for sets of cardinality of the real numbers (or larger). But that's just my math background speaking. I understand what you mean.
Yes, we're in agreement.
"Dense" is sufficient. (An ordered set is dense if: between any two elements of that set, there is another element of the set.)