I'm not sure how it matters if we're in one. Our goal in all realities should be to break free of that reality's constraints.
On the detection part, I predict we won't figure this out until we start simulating realities ourselves; then for each bug we have to fix, we'll check it against our reality. "A simulation inside a simulation!"
Or we may realize that energy is being supplied to the universe from an external source.
Or we may be able to rowhammer a neighboring simulation (or any other part of the machine really). Of course we might accidentally cause memory corruption and destroy our universe, but we might also get root access! Basically execute an in-universe hack to escape our simulation confines. Fun to think about how this would all seem to aliens in the simulation next door when humanity first fiddles with their data then is able to execute arbitrary code.
If we can detect that we're running in a VM, this by definition means that the virtualization is imperfect. We could use these imperfections both to obtain abilities not normally allowed by the VM rules, or contact either the "supervisor" or "parallel VMs", again, gaining abilities not normally possible.
So finding a reliable proof that our world is a simulation automatically means finding some kind of an escape hatch from it, or at least a peephole to the "outside", which arguably would matter.
If we're in a "simulation", ie, in a computer program, there is a possibility we may find a bug, an exploit, or a backdoor, that would let us become "superuser" and do magical god-like things (and most probably, just crash the process, destroying the universe and all there is inside it).
On the other hand, simulation or not, this doesn't exclude or prove the existence of God. A simulation could be generated by unconscious physical processes in the super universe, as they could be created by "people" in that super universes (ie. gods). If the universe is not a simulation, it can still be a creation by a God, or happen to be by itself (or some unconscious meta-physical process).
And as you note, it's also orthogonal to the question of whether the universe is a closed system or not. (both information-wise and energy-wise).
On the detection part, I predict we won't figure this out until we start simulating realities ourselves; then for each bug we have to fix, we'll check it against our reality. "A simulation inside a simulation!"
Or we may realize that energy is being supplied to the universe from an external source.
Or we may be able to rowhammer a neighboring simulation (or any other part of the machine really). Of course we might accidentally cause memory corruption and destroy our universe, but we might also get root access! Basically execute an in-universe hack to escape our simulation confines. Fun to think about how this would all seem to aliens in the simulation next door when humanity first fiddles with their data then is able to execute arbitrary code.