UPDATE!

I have sent a few emails out to friends and have been directed back at an old professor of mine who studies star formation at cosmological timescales, but they are very busy and I don't know if I will hear back from them in a reasonable period that will allow a response in this thread to be seen.

Been really busy recently between dealing with peer review of a paper and TAing a course and trying to keep various projects in my PhD going, but in between I've been looking around. My summarized conclusions follow:

It would appear that the papers linked to use a set of somewhat uniformitarian assumptions about galaxy formation and history that are not necessarily accurate. Most importantly, it assumes that ALL star forming gas within a galaxy's dark matter halo will eventually collapse, accrete, and form stars. This assumption appears to not be a good one.

When you look out into the universe, the vast vast majority of elliptical type galaxies are very red and are not forming stars, whether they have an internal reservoir of gas within their dark matter halo or not – see the link I posted in my other comment to this comment. Grand spirals are mostly forming stars at a steady clip, with only a few tapering down and turning 'green' or eventually 'red' from their initial 'blue' status. Recently, a project called GalaxyZoo which has automated and crowdsourced the analysis of huge numbers of new galaxies observed in the Sloan Digital Sky Survey has taken a very quantitative look at star formation across galaxy types in the universe, and come up with some striking conclusions:

http://blog.galaxyzoo.org/2014/02/21/the-green-valley-is-a-red-herring/

http://arxiv.org/abs/1402.4814

http://arxiv.org/abs/1501.05955

These studies were able to get more information than the instantaneous rate of star formation, and look back along the history of the galaxies by looking at light of different frequencies – huge stars that dont live long make lots of ultraviolet, stars like our sun peak in the green light, while long lived stars peak in the red. They were able to see that among elliptical galaxies, the tiny fraction that are star-forming mostly show evidence of recently being involved in mergers, and that all those that are red and green colored show spectral patterns indicative of very rapid shutdown of star formation, faster than can be accounted for by star formation eating up available gas. They call this fast star-formation shutdown 'quenching'. Something about their formation, either primordially or via mergers of spirals, puts their gas into forms that cannot form stars. The prime suspect is the initiation of regular energetic outbursts from their large central black holes, heating the gas and rendering it too turbulent.

This actually dovetails interestingly with another problem in astrophysics: the 'cooling paradox'. In short, about 90% of the baryonic mass of the universe is in the form of X-ray hot gas clouds blanketing entire galaxy clusters (largely outside the dark matter halos of individual galaxies). This gas is ridiculously thin and immensely hot, and radiating energy rapidly in the X-rays. It turns out that when you figure out how much mass is in these gas clouds and how much energy they are radiating in the X-rays, they should cool and sink down to the centers of the clusters on a timescale of gigayears, probably turning into cool gas flows onto the large galaxies at the centers of these clusters. But they don't. Looking back in time across the universe they are at more or less the same temperature now as they always have been and never seem to cool despite the fact that they are radiating energy. In recent years, for various reasons (images of turbulence in the gas, calculations of the available energy) the prime suspect for the energy source keeping these gas clouds energized has become supermassive black hole jets.

Anyways, as for spiral galaxies, they were able to model the distribution they saw (most of which are forming stars at a steady rate, some of which are tapering off, and some of which are red and dead) as a mixture of populations. One population is forming stars at a steady slowly decreasing rate, much like ours. Another is quenching on a much slower timescale than ellipticals, indicative of a cut-off of gas inflow into their star-forming discs and star formation then slowly depleting their reservoir of gas over a 1-2 gigayear timescale, likely caused by events in their immediate galactic neighborhood disrupting the inflows of cool gas within their dark matter halos onto their star forming discs.

So, when the originally posted paper notes that in the Andromeda/Milky Way system our star shows up as in the 39th percentile, that is assuming that ALL gas in both galaxies will form stars either before or after they merge, when that is not necessarily a given. There is a very good chance that the merger will produce an ellipical with a very rapid quenching period indicative of processes going on that put the star forming gas into a form that cannot form stars. As such, we are almost certainly after the 39th percentile, by a quantity I am unprepared to address (though doing some naive by-hand projections of their curves, I think it winds up being over the 50th percentile).

Similarly, when they claim that we are in the 8th percentile of planets formed universally, it is a bit off base. It assumes all gas in galactic dark matter halos will eventually form stars. A very large fraction of galaxies are large ellipticals which seem to not form stars, no matter their gas status. Of the spiral galaxies available, a subset are having their star formation quenched on gigayear timescales by processes relating to the prevention of gas inflow. And those that undergo mergers (a constantly increasing number) will almost all cease star formation in a rapid quenching very shortly after merging.

The literature I have talked about in the past, looking empirically at star formation rates across time in the universe, have seen the results of these quenching events in the rapid shutdown of star formation starting 11 gigayears ago and continuing to this day. Non-quenched galaxies continue forming stars (less vigorously than early in their lives but still) but a larger and larger fraction of galaxies are quenched, their gas prevented from cooling and condensing to form stars. I am unprepared to extend the analysis of our place in the planet-formation order to the universe as a whole other than to note that we are probably somewhere between the 33rd and 66th percentile of our galaxy, a fairly typical large star-forming spiral. I also note that figure 2 (top right panel) of the original "on the history and future of cosmic planet formation" paper shows a universal rate of terrestrial planet formation that looks rather like a skewed sigmoid, with Earth coming after the period of most rapid formation but the rate leveling off slower than it ramped up, rather than a curve that will continue to >10x as high a number of planets formed as existed when Earth was formed. It seems to me that our position in time and in star-order is unremarkable.

The original paper that sparked this discussion uses a number of simplifying assumptions that don't seem to carry over to the empirical studies; I think the empirical studies are more useful even if the exact mechanisms of galaxy star formation quenching and gas inflow prevention are still kind of up in the air with a few popular frontrunners.

EDIT: There is also the question of what model to use for the production of Earthly terrestrial planets, and if these are likely to be found in giant ellipticals versus spirals. The former contributed much more star formation to the early universe than to the current era, and these stars being unsuitable would explain us showing up a bit late as revealed by some of the purely empirical data I've seen. The paper that started this discussion did take this into account in their galactic and universal curves, with an absolute minimum metallicity cutoff for terrestrial planet formation and assumed that formation was equally likely for stars of all metallicities above that. I don't think anyone really understands this aspect terribly well...