Question on Newton's first law

Very good question. You would do many experiments with many different bodies and you would find that, for all of them, you would have to account for a mysterious force given in terms of a constant vector $boldsymbol{f}_i=-m_iboldsymbol{A}$ that must characterize your system kinematically, because it affects all moving bodies equally in proportion to their inertia. Your version of Newton's law would be, $$ m_{i}boldsymbol{a}_{i}=boldsymbol{F}_{i}-m_{i}boldsymbol{A} $$ You would conclude --or it would be reasonable to conclude-- that $boldsymbol{A}$ is something kinematical characterizing your frame.

This is a deep question.

"But how can we be sure that the fact that an object, with zero net force on it, is accelerating because we are in a non−inertial reference frame and not because of the fact that we haven't accounted for some "real" (as opposed to those called "fictitious") forces."

You can not be sure, as you can also not "prove" physical theories. They either explain your experiment or not, so when you set up an experiment and you measure the trajectory of e.g. a particle not being a straight line, then, according to newton, you have to have either force acting on it or be in an accelerated frame. Now you could study the trajectory of the particle and see whether it is explainable by some force we know, or by some accelerated frame you're in or both together. I think the point is that you either find some explanation in terms of known quantities (forces or accelerated frames) or not, but you can not find out the "truth" because I could imagine that whenever you have an particle flying in some accelerated trajectory through some force, you could also have a particle which flies in a straight line, but you get thrown around such that the particles trajectory appears to you as if it would when the particle itself would be accelerated. I have the impression that it boils down to some form of the equivalence principle, but I can't articulate that right now exactly