Thank you for teasing out these issues.
Some thoughts that came up... As regards why 10 gram x 1 cm / sec is the same as 1 gram x 10 cm / sec, the "conservation of momentum" is a consequence of the symmetry in space, in that the outcomes don't depend on any particular coordinate in space, as per: https://en.wikipedia.org/wiki/Noether's_theorem Emily Noether was an inspiring woman mathematician.
"Conservation of energy" is a consequence of the symmetry in time, in that the outcomes don't depend on any particular time coordinates. So the difference between space and time may perhaps be thought of as the difference between momentum and energy.
Energy also gives an example of what "second squared" can mean. I just wonder what energy means. The Wikipedia article on energy is mystifying: https://en.wikipedia.org/wiki/Energy Mathematically, kinetic energy K = 1/2 m v**2 is the integral (by velocity) of momentum M = m v. Velocity is the expressed relationship between space and time. The higher the velocity, the more weight is placed on time, and the lower the velocity, the more weight is placed on space. Kinetic energy is what it takes to go from velocity v = 0 to v = V. Potential energy is the background energy that explains for us that total energy is conserved.
Perhaps one difference in interpreting the fourth dimension is whether time has a plus or a minus sign. In Minkowski space, used by Einstein, time and space have opposite signs: https://en.wikipedia.org/wiki/Minkowski_space Whereas in Euclidean space/time I imagine they have the same sign. I'm curious to learn more about the reason for that distinction.
Joseph, I like very much your emphasis on units. I found that helpful in learning physics and as a tutor I developed some general principles for my students:
"Every answer is an amount and a unit" (3 is not an answer but rather 3 feet, 3 seconds, etc.) But then (by sleight of hand) a number can become a unit: 3 millions, 3 sevenths, etc.
I'm wondering about the purpose of this breakdown. It's probably partly to distinguish between what we attribute to our mental world (the amounts) and to the physical world (the units). And perhaps it helps us distinguish between answers and questions. Answers are fixed and so they are "contravariant": if we divide up the units by 1000, and go from kilograms to grams, then we have to multiply the amounts by 1000. Whereas questions are not fixed and they are often phrased in terms of (1/unit) as "per unit": How many miles per hour? And if we divide up the hour into 60 minutes, and we get "per minute", then we have to divide up our amount (How many) by 60. I'm just thinking out loud.
I taught that "You combine like units", (to calculate), for example: 3 sec + 2 sec = 5 sec but 3 sec + 2 feet isn't anything (to combine) 3 million + 2 million = 5 million 3 sevenths + 2 sevenths = 5 sevenths 3 X + 2 X = 5 X but 3 X + 2 Y doesn't combine
"You list different units" (to make your answer easy to understand) the marathon was won in: 2 hours + 12 minutes + 8 seconds
You convert different units to same units in order to combine and, in general, to make it simpler to calculate.