To 1.: If, as in a gauge theory, all observables of a theory are invariant under some continuous symmetry, there are unobservable degrees of freedom - there is no way to observe the variables that change under the symmetry.
For a simpler example, suppose you can observe only length $|x_i-x_k|$, not position, then position is unobservable since $x_i\to x_i+s$ is a continuous translation symmetry preserving length. This is in fact the case in real physics - we observe position only since the state of the universe has broken translation symmetry so that we know a few rigid points relative to which we can deduce position by means of computing from measured lengths.
Thus the Poincare symmetry of a QFT has the same status as a gauge symmetry - it is due to unobservable degrees of freedom. Only since we pretend that position has an absolute meaning (in a laboratory frame where experiments testing a QFT are performed), translation and rotation symmetry seem to have a different status.
In general relativity, this is more pronounced, as there not only translations but all volume-preserving diffeomorphisms are continuous symmetries - which in some formalisms is considered a kind of gauge symmetry.
To 2.: A large gauge transformation is local only in a restricted sense. It is nonlocal in that it affects more than any compact region. This makes it large. It has nothing to do with changing boundary conditions. Indeed, boundary conditions equivalent under large gauge transformations are physically indistinguishable. Precisely this causes problems that must be addressed when discussing infrared effects.
To 3.: your Lagrangians as written are the same. To turn a field theory into a diffeomorphism invariant theory (which, according to 1. above, makes it gauged), you need to be more careful. See the entry ''Diffeomorphism invariant classical mechanics'' in Chapter B8: Quantum gravity of my Theoretical Physics FAQ.
To 4.: The Maxwell equations are described in terms of a closed 2-form, hence carry with them a connection (and hence a gauge field), even if this is not explicit in the formulation. That the gauge field is real can be seen from the Aharonov–Bohm effect, which cannot be explained by Maxwell alone since the topological information in the gauge field is lost. And the other interactions need nonabelian gauge fields, where it seems impossible at all to eliminate the gauge fields. (In you final sentence, an ''is'' is missing.)