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There are multiple sensible definitions and treatments of manifolds with corners and their transversality. For my own applications (geometric homology and cohomology, with Friedman and Medina) the best is by Joyce - "On manifolds with corners"
As Will points out below, coproducts of sets are not less common. I've thought about the algebra/ coalgebra question and think of the answer here as algorithmic. To specify an algebra means giving a rule to produce something from two things. To specify a coalgebra means creating a "lookup" function - e.g. find all occurrences of 36 in the multiplication table. I'm no computer scientist, but I conjecture that if this can be made well-posed the lookup question will be more intensive.
Yes. First, I should have elaborated that the higher torsion is all determined by a result treated very briefly on pages 48-49 of Cohen-Lada-May's book. What I was working on before, but have paused that project, is a conjecture that in cohomology higher torsion all arises from divided powers operations. For $S_4$ this means that there is 4-torsion in degrees which are multiples of 4, powers of $y^2$ using notation from the question above. (E-mail me if you want to know more.)
Infinity categories have resolved questions that predated the theory, namely giving a second, streamlined construction of topological modular forms and are the right context for some TQFT's, in particular as given in the Cobordism Hypothesis.
Ben Walter wrote his thesis on Goodwillie calculus in rational homotopy theory. It doesn't exactly get at the questions you are asking, but it does many related things. It provides an explanation for a sort of converse of 1, namely seeing that the derivatives of the identity are (shifted) coLie is immediate in the Lie model for rational homotopy types.
A class of vector bundles I like to consider for intuition are flat bundles, of the form V \times_G E over E/G = B say for discrete groups G. Here as one goes around a loop in B whose endpoints differ by the action of some g when lifted to E, the bundle will be "twisted" by the action of g on V. When V is one-dimensional, this will be a number. For the tensor product of such line bundles these numbers are multiplied. My intuition more generally is "product of twisting" but of course that should be taken with a grain of salt.
One of my favorite computations is that the inclusion of $(C_2)^n$ in $S_{2^n}$ as a transitive subgroup (by acting on itself) induces a map in group cohomology whose image contains the Dickson algebra. The argument is akin to standard embedding of cohomology of $BO(n)$ as symmetric polynomials. It is relatively elementary and uses many standard properties of characteristic classes. It is due to Madsen-Milgram (and is recalled in a joint paper of mine with Giusti-Salvatore).
This is my favorite way, as this count is essentially a geometric representation of the first cohomology of symmetric groups with Z/2 coefficients. The entire cohomology can be understood in similar terms!
