Three points is the minimum necessary to fit three parameters, so as long as nothing is degenerate, you should be fine. Now that being said, fits to nonlinear models can be tricky, because you can get stuck in local minima, but there are techniques for avoiding that.
That the model can perfectly fit any data is a *problem* if we're asking, as this paper is, whether the model describes reality or not! It's not falsifiable!
Yes, it's falsifiable. If you cannot fit the model to the data, then the model must be wrong. That's the direction science technically works, but almost never how it is reported in the media. The fit here is considered a good thing, but cannot conclusively prove the model (as I think you are saying).
In principle, you could fit every single atomic/molecular spectrum to the data and falsify everything else based on some goodness of fit test, but no one does this.
Another thing about these molecular spectra is this: if there is an emission line missing, then it cannot be the spectrum you are fitting. These things do not change with environment, and can only be red or blue shifted due to special relativity. Every PAH spectrum must have the same lines. It's like a fingerprint.
(I'm aware that the emission spectrum for (their specific chosen model of) PAH dust has a fixed shape. That's not my confusion).