One last paragraph on demographic history and likely future trends, and then let's leave it. I am making two points, both of which are clearly substantiated by the historical record. First, the current fertility levels below replacement level in the developed world are not unprecedented. Second, attempts to extrapolate fertility trends into the future have always turned out to be flat wrong. Neither of us will be around in the year 2150 to see which of us is right, but fertility never has been stable at low levels for long periods of time, and there is no reason to suppose that it will be now. To rebut the details of your demographic history, fertility fell right through the 1920s to a low in the mid-1930s, not just in the Great Depression; the 1920s were not great economically, but not disastrous, yet fertility fell. No clear reason there.
Reproduction rates are not far lower now than those experienced in the 1930s: For the United Kingdom, the Net Reproductive Rate is higher now than in the 1930s, and in the United States the level now is about the same as it was in the 1930s. Fertility decline came a bit later in Spain and Italy, so their fertility was higher in the 1930s than now, but I believe that their current very low fertility is in part a period phenomenon, affected by changes in the timing of births rather than the ultimate quantity of births. Rates have stayed low longer now than in the 1930s, but to describe the current fertility situation as a free fall is ludicrous: Fertility is higher now in the United States and the United Kingdom than 20 years ago, by a margin of almost 20 percent in the United States. Going up is not a normal characteristic of a free fall. All now-developed countries experienced fertility declines in the early part of this century, though with some (surprisingly small) differences in timing, such as those noted above for Spain and Italy. Fertility decline in the developing world is new, however: Fertility was probably rising in many parts of the developing world until 1960, certainly in terms of the NRR as mortality declined sharply.
So, I'm not arguing that another baby boom is just around the corner. I don't know what is around the corner, you don't know what is around the corner, the United Nations doesn't know what is around the corner. So scary stories about the implications of a 3.6-billion Low Variant population in the year 2150 are just plain silly (and not very scary--3.6 billion is still a billion more than the population in 1950).
I agree there is little evidence that rapid population growth influences per-capita income decisively one way or the other. Arguments that resources are fixed, and that a higher population will reduce resources per capita, are naive, because resources aren't fixed. On the other hand, arguments that higher population density improves economic performance by increasing opportunities for specialization or increasing the rate of innovation are not empirically supported either. The National Academy of Sciences report that you cite actually concludes that "[s]lower population growth can be expected to increase the ratio of capital to labor, which in turn will increase the level of per capita income," even though it goes on to say that the effect may be relatively modest. Increases in human capital through better education, driven by improved returns to education, may magnify the effect and have a particularly important impact on poverty. So population size, at least around current levels, is close to a wash as far as average economic conditions go, but slower growth can be expected to make it easier to achieve basic needs for the world's poor.
Be that as it may, you admit that population growth cannot continue indefinitely (or even for very long) without reaching limits that the Earth clearly cannot support, so we have to stop sometime. The question is, as you say, when and how.
Now seems an excellent time to stop. Take greenhouse gases. In the long run, we want standards of living to become more equalized around the globe than they are at the moment. Presumably, we want to see this come about by raising the standards of living in developing countries rather than by reducing standards of living in developed countries. Let's assume the LV population projection of 7.7 billion for the year 2050, and look at its implications for carbon-dioxide emissions. Developed countries are about four times as efficient as developing countries at producing a dollar of Gross Domestic Product per unit of carbon dioxide emitted, in part because service industries are not big producers of CO2. However, per-capita GDP is so much higher in the developed world that an equalization of income to developed-country levels in 1990 would produce a huge increase in CO2 emissions, from about 5.2 billion tons of carbon per year to about 18.1 billion tons by 2050. We'll be toast. But we'll be better off than Bangladesh, which will be underwater. According to the LV projections, the population of Bangladesh in 2050 will be 178 million even without global warming, a population density three times that of Rhode Island and one-third that of the District of Columbia. For a community that will still be largely agrarian, that's a lot of people.
In the mid-1990s, more than half the population of the world lived in low-income, food-deficit countries--that is, countries that do not produce enough grain to feed their own populations and are not rich enough to import the difference while maintaining an acceptable standard of living. Population growth over the next few decades will exacerbate this situation. Better farming techniques and greater use of new technologies could increase food supplies in these poor regions, but these techniques and technologies are out of reach for subsistence farmers in poor countries. The same is true for tropical forests (and their rich biodiversity). Improved farming techniques, alternative fuel sources to wood, and alternative employment opportunities for rural peasants would indeed reduce the destruction of forests, but these alternatives cannot be implemented in poor countries. Having observed Central and Eastern Africa over the last 30 years, there is no question in my mind that population pressure has contributed to widespread destruction of forests and wild areas. Other factors, often inappropriate subsidy or tax programs that distort incentives, undoubtedly play a role too: Good governance can compensate for population growth, and reduced population growth can compensate for poor governance.
I do not personally believe it will, but let us suppose that the LV does come about in what you call "the West." First, your view of "the West" as "the best hope that mankind will move forward" is, to say the least, ethnocentric. It is less than 60 years since "the West" was convulsed by a degree of brutality matching anything that Cambodia or Rwanda has produced. It is less than five years since Bosnia was gripped by brutal ethnic strife. Centuries of civilization have produced a very fragile product.
Second, what would be the effects of an aging population? You ask where the money will come from to support the elderly and suggest that it will come from higher taxes or lower benefits, with a cut in the standard of living. Here it is necessary to consider dependents in general, not just the elderly. The effect on the overall dependency ratio, the ratio of persons under 15 or over 65 (loosely "dependents") to those aged 15-64 (loosely "productive") between now and 2050 is a modest increase: from 0.49 in 1995 to 0.69 in 2050. The reason why the change is modest is that as the proportion over 65 increases, the proportion under 15 declines. If the population of those who are 15-24 is included as dependent (as it probably should be in a society where tertiary education is the norm), the change is even smaller, from a dependency ratio of 0.89 to a rate of 1.00. Comparable figures for 1960 are 0.58 (using those who are younger than 15) and 1.08 (using those who are younger than 25). Clearly the dependency implications of the LV for the West in 2050 are not unprecedented. Even if this were not the case, the population is getting healthier and living longer. If there is really a shortage of productive labor, real wages will rise, and three things will happen: First, current high unemployment rates (in Europe) will fall, which is good; second, people will be able to work to later ages, which is good or bad depending on personal preferences; third, immigration into the "mature" populations will increase.
So is immigration good or bad? It is good for the immigrants, and I think we can assume that if it is clearly good for the existing population, they will also be in favor of it. Immigrants are predominantly young adults, so they reduce the dependency ratio sharply. The costs of raising children are borne by the sending population, whereas the labor-force benefits accrue to the receiving community. Immigration is a very smart way of taking advantage of other populations, and has played an important role in fueling U.S. economic growth this century.
So now let's turn to your familiar arguments about extra sadness, missing children, and loneliness. Personal control over childbearing is greater now than ever, even to the extent of giving birth in one's 60s. Thus people are better able than ever to have the number of children they want. In these circumstances, it makes no sense to talk about missing children: If people wanted children, they would have them. Having them when not wanting them would presumably contribute to extra sadness. What about loneliness? People can be lonelier in New York City than in a hamlet in rural Kansas, so loneliness is not reduced by population density. We value family ties, but we also value friendships. Further, the nature of family structures depends not on averages (such as 1.4 children per couple) but on the distribution: It could be that the average of 1.4 would be achieved by about half the population (by choice) having no children and the other half having three, in which case all the children would have two siblings, aunts or uncles, etc.
You conclude, "We ought to make it easier for young people who want to have children to have children." I would argue that it has never been easier for young people to have children: Sexual mores have relaxed, ready access to contraception and abortion make choice easy. Should we encourage young people to have children? Here, I would argue that human-capital formation (education) is the most important component of increased human welfare, contributing to, but far eclipsing, per-capita income, and that childbearing should be postponed until such formation is complete. Perhaps you could describe the barriers you see to young people having children, and explain how they should be removed.