Although to you, category theory is merely an inefficient framework for data about logic and programming languages, to mathematicians working in areas like algebraic geometry and algebraic topology, categories are truly essential. For us, some of our most basic notions make no sense and look extremely awkward (in fact, some of them are from pre-category days, and no one really knew what we wanted intuitively until after we started using categories) until you phrase them in terms of categories and universal properties of objects and morphisms (and 2-morphisms, etc). Additionally, it helps us tell what various types of mathematical objects have in common, and how they relate via functors and natural transformations.
As far as recasting algebraic topology and algebraic geometry in terms of lambda calculus, I'd rather like to see that if anyone can manage to, say, give an alternative definition of stack or gerbe or model structure which are more intuitive without using categories and groupoids and the like.
There are a lot of questions here, but I'll try to answer them all.
Should every mathematical theory take place in a ∞-category? Or is 'real' mathematics basically evil?
I would say that all mathematics should take place in its natural context. Sometimes you have things that are sets where equality makes sense, like an ordinary presheaf, and then you work in a 1-category. Sometimes you have things where only isomorphism makes sense, like a presheaf of categories, and then you work in a 2-category. Etc.
It is true that any n-category for finite n can be considered a special case of an ∞-category with only identity cells above n, so in this degenerate sense all n-categories are ∞-categories, and thus one might say that "all mathematics takes place in an ∞-category" — at least if one believes that all mathematics takes place in an n-category for some n! But even that is not clear, e.g. some mathematics naturally takes place in other categorical structures, such as a double category or a proarrow equipment. Some mathematics uses no category theory at all (at least as far as anyone has noticed so far), and so it would be a stretch to say that it takes place in any sort of category.
Anyway, may we think of it as a usual functor, without turning into troubles? Or is it important, in practice, to have this higher category theoretic point of view? Or is it possible to turn this functor into a honest functor, by choosing the tensor products $M\otimes_A B$ carefully?
I would say qualified yes, yes, and yes, respectively. You can think of it as a usual functor as long as doing so doesn't cause you to think that it behaves in any way that a pseudofunctor doesn't! Which is sort of a vacuous statement, but the point is that pseudofunctors really shouldn't be a very scary concept (as opposed to a technical definition, which might be a bit complicated, though cf. Harry's comment) — they really are just like ordinary functors, except that you're dealing with things (e.g. categories) for which it doesn't really make sense to ask morphisms to be equal, only isomorphic.
On the other hand, the "higher category theoretic" fact that pseudofunctors are not all strict functors is very important. I believe that Benabou, the inventor of bicategories, once said that the important thing about bicategories is not that they themselves are "weak," but that the morphisms between them are weak. In particular, although every bicategory is equivalent to a strict 2-category, not every pseudofunctor between bicategories is equivalent to a strict functor.
But on the third hard, it is true that any pseudofunctor with values in the 2-category Cat is equivalent to a strict functor. In the language of fibrations, this says that any fibration is equivalent to a split one. Tyler mentioned one construction of an equivalent strict functor in the case of modules and tensor products. There is also a general construction which, applied to the case of modules, will replace $Mod_A$ by a category whose objects are pairs (M,φ) where M is an R-module and φ:R→A is a ring homomorphism. We regard such a pair as a formal representative of $M\otimes_R A$ and define morphisms between them accordingly, to get a category eequivalent to $Mod_A$. Now the extension-of-scalars functor $\psi_!:Mod_A \to Mod_B$ is represented by the functor taking a pair (M,φ) to (M,ψφ), which is strictly functorial since composition of ring homomorphisms is so.
Best Answer
Fundamentally I agree with Mike Shulman's comment and I do not really want to claim the following fancy language is at all necessary to answer this question, but you may (or may not) find it illuminating.
From the standpoint of higher category theory, categories (i.e., 1-categories) are just one level among many in a family of mathematical structures. Typically a mathematical object will "naturally" exist as an n-category for some particular n. For example, Set is naturally a 1-category, while Cat is naturally a 2-category. Your examples Series and so on seem to just be 0-categories, i.e., sets, since as Pete explained in his answer, there is no obvious natural notion of morphism between infinite series. Asking why Series is not a 1-category is like asking why Set is not a 2-category; these are just not the natural categorical levels that these objects live at.