F-Droid was offered to be installed as part of 1st boot / installation. That (and Aurora Store) was chosen.
What has been attempted:
Additionally: the non-working F-Droid app shows “Updating repositories” infinitely on the upper screen.
Is there any way currently to get F-Droid store to work?
Did you tried with different preferred network e.g 2G, 3G, 4G?
Does it work with WiFi?
4G tried, 3G not even available in this area no longer (so can’t do that).
Cannot try on WiFi either; i have no WiFi device. Only the mobile data (SIM) and ethernet.
Probably same problem I was encountering with this For me, I have not been able to get any Android app connect to internet via mobile connection
Also if you have your previous phone still, you could probably use that to share the wifi and get at least something installed from F-Droid. Of course if those apps need internet connection to work, then that solution is not that useful
So, it appears that the default installation feature defaults leads to dysfunction. This should be sorted out ASAP IMO. At least the default stuff should work (especially as F-Droid and Aurora Store are checked to be installed and utilized on th get go).
Of course there are countless ways to tinker and “dick around” - but the point is: these are vital features to work OOTB. A regular user will be so confused it’s not even funny. Also myself am not no longer willing to tinker with this stuff anymore; i bought a new an dandy phone and i expect these basic things just to be fixed.
I’m sorry for mentioning the one workaround I’ve found so far. I’ll try to keep in mind to not bother you with any suggestion in the future
Don’t be sorry pal, i knew that “workaround” already. You’re missing the point. And this section is for bug reports.
I know very well this section is for bug reports, as you can see I’ve used it myself for reporting this problem. You are the one who somehow gets offended that I provided the partial workaround I happen to know to a problem, you specifically asked: “Is there any way currently to get F-Droid store to work?”. How I’m missing the point if you reply to my provided workaround directly with complaining to me about tinkering and things getting fixed?
Looky, this is nothing personal. Just tech - a new phone. And no one is offended as you suggest 
all is quite cool --but these things are to be fixed, and that is why we are passionately reporting these here.
Not really a fix, but I’ve been enjoying Droid-ify as alternative F-Droid client.
Much more pleasant and modern UI, more reliable in my opinion. Maybe this one works for you.
Thanks - seems nice! Will check it out as the F-Droid client starts to fly properly (so Droid-ify would update along with anything else then) 
It is the issue @jsyri mentioned. Currently you’ll have to use a WiFi network to install, update and use Android apps - I understand that this is inconvenient, but there’s no workaround or fix yet. It’s a known issue and will be fixed in a later release.
F-Droid and Android Store are installed using Store, which is a native Sailfish OS application, which is why installing those works in the first place.