A horrible book. No wonder that you sound confused to me. So what is this "use of Friedmann-Lemaitre model without using Robertson-Walker metric"? The one of formula (4.23)? In this case, sorry, but this is what is usually named the FLRW metric in other (non-comoving) coordinates. To name this a different metric is confusing. If not, what else?
The additional predictions are claims which can be falsified, by observing what these principles forbid. If the space would be, for example, topologically nontrivial, say, $$S^3$$, this could be observable, at least in principle. Not that this would be easy, but under certain circumstances it would be observable. Thus, in principle it is observable. Which is the point.
And if there is a principle, what matters is not that we like it, or that it has some outcome that we like, but that we have not observed yet any violation of the principle, but, if it would be false, this could be, in principle, observable.
No. We even have strong evidence that in the early universe we have $$a''(\tau)>0$$. Which excludes the simple Friedman solutions of GR. There is no usual matter which could give this.
Ok, inflation theory has managed to construct such matter models, using scalar fields with a change of the vacuum state. But if one allows arbitrary matter models, the Einstein equations of GR predict nothing at all. Indeed, if we observe, instead of the Einstein equations, $$ G_{mn}(x) = T_{mn}(x) + Err_{mn}(x)$$ with some arbitrary symmetric $$Err_{mn}(x)$$, all one needs is to name $$ T^{dark}{mn}(x) = Err_{mn}(x)$$ for some unconventional dark matter and we have agreement with GR. So, GR is a predictive, falsifiable theory only if one adds some nontrivial assumptions about matter fields and their stress-energy tensor. From this point of view, without making theoretical assumptions about the non-existence of certain types of dark matter, dark energy, or other dark things we are never in a situation that we can decide between theories empirically.
No. There are a lot of different vacuum solutions. They have quite different metrics, but the stress-energy tensor of matter is zero in all of them. To illustrate the quality of your book p.157:
The Riemann tensor is $$R_{abcd}$$, the Einstein equations are $$G_{mn} = R_{mn} - \frac12 g^{ab}R_{ab} = T_{mn}$$ which depend only on (and therefore defines only) the Ricci tensor $$R_{cd}=g^{ab}R_{abcd}$$. So the Riemann tensor has much more components than equations to define him, only a few linear combinations are determined by the Einstein equations.