Falsification is, in general, not actually a useful metric, because evidence and strength of belief are quantitative and the space of hypotheses is larger than we can actually scan.

I'd note that the layperson's description of a black hole is, in fact, false. Squeezing a given mass into a singularity doesn't make it heavier. The mass stays the same, but the density goes up. Even as it collapses into a black hole, the Schwartzchild radius will be much smaller than the original object's size - about 3km for a 1 solar mass black hole. If you personally could do the squeezing on a small enough object, what would happen is eventually the object would go from resisting collapse to sustaining collapse, then explode with the light of a billion suns. For a tiny fraction of a second during that process it would leave behind a core that, in an ultramicroscopic volume of space, kept light from escaping.

This actually poses a kind of Gettier problem. If you try squeezing things really hard, and correctly weigh the evidence, you'd decide the theory was probably false. And it is false. But the experiment doesn't prove anything.

The next side to the resolution is: What do you mean the Roman commoner "has a theory"? Where did it come from? Why is he thinking about it at all, or giving it any credence? If alien beings with godlike powers descended from the heavens and tried to explain relativity, but this is what he misunderstood, that's actually pretty strong evidence for both this and the existence of gods! Or if he made it up, how or why is that what he made up?

And of course: sometimes the road to a correct understanding goes through a maze of contradictions and things you don't have any valid frame of reference to interpret. Science and reason don't promise an answer to resolvable questions soon. And there probably are lots of questions that are unanswerable in principle, including sometimes the question of which questions are unanswerable in principle.

Follow the improbability. What drew that particular theory to the person's attention, either the hypothetical Roman commoner or the person arguing that we can't yet test their hypothesis about God? If the answer is "nothing", as is literally the case for the imagined Roman, then we need not concern ourselves further with the matter. If the hypothesis about God is not already entangled with the world, it fares no better.

Technically it's still never falsifiable. It can be verifiable, if true, upon finding yourself in an afterlife after death. But if it's false then you don't observe it being false when you cease existing.

https://en.wikipedia.org/wiki/Eschatological_verification

If we define a category of beliefs that are currently neither verifiable or falsifiable, but might eventually become verifiable if they happen to be true, but won't be falsifiable even if they're false—that category potentially includes an awful lot of invisible pink dragons and orbiting teapots (who knows, perhaps one day we'll invent better teapot detectors and find it). So I don't see it as a strong argument for putting credence in such ideas. 

Sure.  There's lots of things that aren't yet possible to collect evidence about.  No given conception of God or afterlife options has been disproven.  However, there are lots of competing, incompatible theories, none of which have any evidence for or against.  Assigning any significant probability (more than a percent, say) to any of them is unjustified.  Even if you want to say 50/50 that some form of deism will be revealed after death, there are literally thousands of incompatible conceptions of how that works. And near-infininte possibilities that haven't become popular.  Note that if it turns out that consciousness is physical and just ends when the physical support for it terminates, then nobody will be able to observe that.  It's a permanent "no evidence" situation.

All that said, it's hard to argue against someone else's choice of priors (what they believe before evidence becomes available).  Maybe they have access to experiences you don't.  Maybe they weight some kinds of social evidence more heavily (the 'prophets' theory that there are historical or current people with more direct connections).  Maybe they're even right - you don't have access to any counterevidence, right?  By "hard to argue", I mostly mean "hard to be sure yourself", but also literally "not worth arguing".   We'll all find out soon enough, right?

Or maybe it's all relative - it's true for them, and not for you.  Or maybe it's weirder than we can imagine.