1)" But how does the mere fact of a formal mathematical duality prove that the theories are physically identical?" In the formulation of this question there are many strange redundancies. A theory is a mathematical structure. Two theories are the same if there exists a natural identification between the corresponding mathematical structures. A given theory is supposed to describe a given physics. By definition two identical theories describe the same physics and so are "physically identical". In some sense any mathematical result is formal so "formal mathematical duality" should be simply "mathematical duality" or even just "duality" because, as I recalled it, a theory is a mathematical structure and so a duality between theories is obviously "mathematical".
About the Maxwell equations. The electromagnetic duality does not identify the electric and magnetic fields but says that which one is called "electric" or which one is called "magnetic" is arbitrary. Two physicists using both the Maxwell equations but with an opposite convention about what is electric and what is magnetic describe the same physics. The fact that a theory is self-dual does not mean that a sector is redundant. It just means that some labels attached to the theory are in some sense arbitrary. If you think to a theory as a mathematical structure, a self-duality of the theory is simply a non-trivial automorphism. We are not quotienting by this transformation.
2)I am not sure to understand this question. Yes, some dualities only make sense after some compactifications. If it is possible to decompactify both sides of the duality at the same time then we expect the duality to still make sense for the uncompactified theories (the objection about the "topological inequivalence" is in general resolved by some degree of locality of the theories). But in general the point of these dualities is that it is not possible to decompactify both sides at the same time, it is why there are non-trivial, and in this case the question does not make sense.
3)To define M-theory as the "strong coupling limit of IIA string theory" is doing nothing if we have nothing non-trivial to say about this limit. The non-trivial statement is that this theory is 11 dimensional and has 11 dimensional supergravity as its low-energy limit. The question is essentially where does this claim come from. The study of D0 branes indeed plays an important role here. The point is that type IIA string theory contains an bound state of $n$ $D0$-branes for every integer $n$, each one of mass proportional to $|n|/g$ where $g$ is the IIA coupling constant. In the limit $g \rightarrow 0$, this infinite number of states become massless. Do you know a mechanism to produce an infinite number of massless states ? It is not known how to do such thing in a local quantum field theory or in the known string theories. The only known mechanism to do that is a Kaluza-Klein compactification in the decompactification limit.
In fact there are other independent arguments in favour of this 11 dimensional picture. For example the fact that the low energy type IIA supergravity is the dimensional reduction of the 11 dimensional supergravity and that in this identification the value of the dilaton is directly related to the radius of the 11 dimensional circle. Furthermore the IIA/M relation is only one small piece of the full web of dualities describing the strong coupling dynamics of the various string theories. For each of them one can give non-trivial arguments and what is remarquable is the consistency of the global picture.
4)The fact that "a weak-field theory limit has a myriad of completions" is just not correct in general. Even in standard quantum field theory, to find a UV completion of an IR effective field theory is already a non-trivial problem which seems to have no solution in Lagrangian terms in spacetime dimension $d>4$ and requires non-abelian gauge fields in $d=4$. But it is true that once these restrictions are understood there can be many possible choices. But it is certainly not what we know for theories containing gravity. To find a UV completion of an IR field theory containing gravity is just difficult and before the proposal of the existence of M-theory the only known examples where string theories. For example the only known quantum theory whose low energy limit is type IIA supergravity theory is type IIA string theory and the fact that it is possible to UV complete type IIA supergravity is a kind of miracle from the field theory point of view: it seems that one needs string theory to do it. It is why many string theorists thought in the 80s that 11 dimensional supergravity was just an ordinary non-renormalizable field theory without UV completion. It is the arguments given in 3) which suggests the existence of a UV completion of 11 dimensional supergravity and we call this theory M-theory, which does not seem to be a local quantum field theory or to have a perturbative string theory description. If there were many known quantum gravity theories in 11 dimensions, we could ask which one is the strong coupling limit of type IIA string theory and so should be called M-theory. But it would be an embarassement of riches and we are not in this situation. Rather the fact that we have arguments showing that the strong coupling limit of type IIA theory is 11 dimensional implies the non-trivial fact that there exists a 11 dimensional quantum gravity.
There is no expected problem of commutativity in the double limit weak energy/compactification simply because the Kaluza-Klein modes are massive (because we are compactifying on a circle of given radius. Things are different for more general compactifications on possibly singular spaces) and so do not affect the low energy dynamics. I don't know what is the needed "very exotic handling in a 11D membrane theory". As the membranes are massive, they do not enter in the low energy discussion anyway.
The fact that at low energies M-theory reduces to 11 dimensional supergravity is imposed by supersymmetry. 11 dimensional supergravity is the only field theory with at most two derivatives in the Lagrangian with $N=1$ $11d$ supersymmetry, as the type IIA supergravity is the only field theory with at most two derivatives in the Lagrangian with $N=(1,1)$ $10d$ supersymmetry.