Here's the open access article on arXiv: https://arxiv.org/abs/1711.109... I don't think they determined the statistical significance

At a redshift of 7 the universe is about 770 Myr old (with either Planck or WMAP 9 cosmology) which is close enough to 1 Gyr that you can say "about a billion years" (I guess it sounds better in a pop. science article).

I have more of a problem with the line "there were no galaxies or stars to illuminate the heavens". They existed, it's just that the universe was opaque during this period.

This is because it's rarely done under the banner of atheism. It's also worth noting that there are many more religious people than atheists in the US so they have more resources. There's a discussion on the topic here.

To directly answer the question, here are some examples of secular organisations helping the homeless:
Feeding America
The aliveness project

There may be religious people in the organisation, but it's not a religious organisation. It's not always the same the other way around: two atheist groups were prevented from helping out in religious soup kitchens this year.

Actually antitheist has now come to mean, at least in the secular crowd, one who is opposed to the idea of religion and gods, i.e. in the usage of Hitchens: "I'm not even an atheist so much as I am an antitheist; I not only maintain that all religions are versions of the same untruth, but I hold that the influence of churches, and the effect of religious belief, is positively harmful."

We might have a chance once the >30m telescopes start coming online (e.g. ELT). Up until then there's no chance, given these planets are at ~150-300 AU (5-10 times the radius of Neptune, who's orbital period is ~160 years) and therefore have orbit periods of thousands of years.

Yes, according to the arXiv article. It's located in a region of recent star formation.