- AFAIK, people have not spent much time formalizing Wightman-style axioms for QFT in a category framework. On the other hand, categories and functors have been essential elements in formulating algebraic QFT on curved spacetimes. To translate from an algebraic to a Wightman formalism, one takes field operators and computes their n-point correlation functions with respect to an extra datum of a vacuum state. In particular, an algebraic QFT is defined as a kind functor between a category describing spacetime regions and a category of algebras (of "quantum observables"). Of course one can collect such functors into their own category with natural transformations as morphisms. Whether this is a useful thing to do is of course to be decided by every practitioner for themselves. Here's an example of a work where the categorical framework is put to good use (a tough read if you go in cold, but a good source of backward and forward references):
Fewster, Christopher J., Locally covariant quantum field theory and the problem of formulating the same physics in all space-times, Philos. Trans. A, R. Soc. Lond. 373, No. 2047, Article ID 20140238, 16 p. (2015). arXiv:1502.04642 ZBL1353.81088.
To be fair, the factorization algebra approach to QFT, which also of course uses categories, is an intellectual descendant of the Euclidean version of the Wightman axioms. A comparison to the two categorical frameworks can be found in the works of Gwilliam and Rejzner arXiv:1711.06674 and arXiv:2212.08175.
- A Wightman or algebraic axiomatization of QFT does not refer to any action functionals, so that part of question 2. is ill-posed. From the algebraic side, two algebras of observables $A$ and $B$ gives rise to the algebra of observables $A\otimes B$ which unites both sets of degrees of freedom of both systems. This operation can of course be lifted to the level of the functorial definition of QFTs. Once translated to the Wightman language, it corresponds to the remark in the comment of James Hanson: joining two sets of quantum fields without interactions. When it comes to starting from an action functional one can in principle formalize the construction of corresponding free or perturbatively interacting QFT in your favorite axiomatization, which may or may not involve a categorical framework (again any benefit or down side is up to the practitioner to decide). The state of the art (from a few years ago) for turning action functionals into perturbatively interacting algebraic QFTs on curved spacetimes is recorded in
Brunetti, Romeo (ed.); Dappiaggi, Claudio (ed.); Fredenhagen, Klaus (ed.); Yngvason, Jakob (ed.), Advances in algebraic quantum field theory, Mathematical Physics Studies. Cham: Springer (ISBN 978-3-319-21352-1/hbk; 978-3-319-21353-8/ebook). xii, 453 p. (2015). ZBL1329.81022.
