Genady

Calculus is a part of mathematics. Any more meaningful explanation would require some mathematical education. Maybe you could point more specifically, what don't you understand or what do you want to know?

I don't know why you say and what you mean saying that "space expansion is the only interpretation of cosmological redshift." Thus, I don't know what you are asking to justify. This is up to moderators.

I am not justifying it. I am answering your question, "it is called space expansion and not spacetime expansion. Why?" As I've said above, if you wish to suggest a different interpretation, go ahead.

In the metric, the second term on the right is spatial interval. As you see, it depends on the scale factor, a(t), which can change with time. OTOH, the first term which is temporal interval, does not change.

If you want to suggest a different metric for the homogeneous and isotropic universe, go ahead. If you want to suggest a better model of the universe, go ahead. Anyway, there is only one metric, the spacetime metric. In the FLRW metric, the space rather than the metric of space is expanding.

In the second part of the first sentence that you've quoted in the OP. It applies to any redshift, the cosmological one included.

We use GR. It is both. This how the metric of homogeneous and isotropic universe works. See FLRW metric.

A string is wound evenly around a circular rod exactly four times. What is the length of the string if the length of the rod is 12 cm, and its circumference is 4 cm?

Yes, relativistic Doppler effect includes time dilation. As I know of a solution to the twin paradox that does not include relativistic Doppler effect, could you refer to a solution that does?

PS. The mention of "always" in the OP means, "while the train moves forward."

That's the only part I know of. +1

Stew of Everything:

When a train moves forward there is always a part of it that moves backward relative to the ground. What part is it?

Yes, with a little subtlety. In democratic societies there are often several versions of history, accepted by different groups.

Perhaps you know that it is very difficult to modify constitution. Unfortunately, people go with it.

Constitution is a law that applies today. Unlike the historical facts.

Because people don't care?

No, not only. You use decimal system here, in your "essential step":

I think that if not for the continent underneath, the ice cover would be even more symmetrical.

Sorry @Mordred, I can't participate. I'm not trained on this level.

It is a Prime Law that all advanced civilizations use ten fingers for counting.

Oh, this is a good trick! +1

Let's go to the next plank under the top two. To stick out without falling these two can have their center of gravity on the edge of the plank under them: This position adds another 0.25m making x = 1 + 1/2 + 1/4 m. How far can we go if we continue like this?

And you are right. +1

OK.