Short answer:

It's reasonable to ask "how do we know what we know about historical demography?", and the answer is "by examining many kinds of evidence very deeply."

Discussion:

Wikipedia articles can be good or bad -- and we don't use them as citations here-- but at least some of sources that are being cited there are first rate; you don't seem to have read them.

So you ask

For classical Rome, they say data is lacking but 'computer models' suggest... . Computer models, based on what exactly? How do you know that these computer models are correct?

. . . but you didn't read the source they cited. The source they cite for that is

Frier, Bruce (2009). "Chapter 27: Demographics". The Cambridge Ancient History XI: The High Empire, A.D. 70–192. Cambridge University Press.

which is itself a kind of review article with very deep sourcing. If you want to understand "what's known of Roman demography", read that article, rather than the Wikipedia piece, which is effectively a tertiary source.

I can also recommend Walter Scheidel's chapter on demography in The Cambridge Economic History of the Greco-Roman World (Cambridge University Press:2007) worth reading alongside Frier, different scholar covering essentially the same terrain.

I can add a bit about the many sources. We have tremendous amounts of data from the Roman empire, lots of funerary monuments, lots of written records, even a few census returns. We've got the Romans own actuarial tables, estimating the value of an annuity, this reflects how long the Romans expected a person to live on average, from a given age. This calculation has been done, here:

Frier, Bruce. “Roman Life Expectancy: Ulpian's Evidence.” Harvard Studies in Classical Philology, vol. 86, 1982, pp. 213–251.

Computer modelling is done for things like population pyramids-- if we see evidence of, say, ten 80 year olds, what does that imply about the number of people in that generation? Given expected mortality rates, you can infer a likely range for the number of 50 year olds there were thirty years earlier, in order to get to ten surviving 80 year olds. How these issues work in a Roman context is discussed here:

Scheidel, Walter. “Roman Age Structure: Evidence and Models.” The Journal of Roman Studies, vol. 91, 2001, pp. 1–26.

The basic statistics of demographic modeling that's often done for the Roman population was developed by Ansley J. Coale and Paul Demeny in the 1960s-- a discussion of the Coale-Demeny formulas and how to interpret necessarily incomplete historical data in them is itself a very big topic, see

Ansley J. Coale, Paul Demeny and Barbara Vaughan. "Regional Model Life Tables and Stable Populations" (Elsevier:1983) for a complete description of the demographic models, how they're arrived at and how they conform to observed data.

There is a very long standing scholarly inquiry into the size of the Roman population -- which includes estimates of the life expectancy, necessarily. This starts with the work of Karl Julius Beloch in the 19th century, and it's field to which many scholars and more data has been added with each generation, see:

Bowersock, G. W. “Beloch and the Birth of Demography.” Transactions of the American Philological Association (1974-), vol. 127, 1997, pp. 373–379.

Lo Cascio, Elio. “The Size of the Roman Population: Beloch and the Meaning of the Augustan Census Figures.” The Journal of Roman Studies, vol. 84, 1994, pp. 23–40.

Note that Beloch was particularly interested in the figures in the celebrated Augustan census mentioned in the Res Gestae, figures which have been the subject of a lot of subsequent analysis.

Elsewhere on this reddit, we've had a lot of discussion of Roman legionaries, and their pension, and the likelihood that they would live long enough to collect it. Some of the sources referenced in these threads include:

Brunt, P. A. “Pay and Superannuation in the Roman Army.” Papers of the British School at Rome, vol. 18, 1950, pp. 50–71.

and

Scheidel, Walter. Marriage, families, and survival in the Roman imperial army: demographic aspects in "The Blackwell Companion to the Roman Army" P Erdkamp (ed.), Blackwell: Oxford and Malden, 2007

So that's just a starting point, a look at some of the sources for just one of the periods you're asking about. Understand just how big a subject this is-- the Wikipedia article is a very thin gloss on a genuinely massive subject, it is notably incomplete. They have, for example, just one reference to effectively a tertiary source for one period in China . . . to read comprehensively in Chinese historical demography is a lifetime's work.

So start by picking "which particular civilization do I want to learn about" -- and then start reading. If the Wikipedia article has made you curious and pointed you to some sources, that's good . . . but that's about all its good for.