I find the area your question covers very interesting and am looking forward to seeing what answers people come up with. I thought I would supplement my rather minimal answer from a few days ago. I'm not really providing any work that links all these ideas together, rather I'm giving some useful references for your quest.
Peter Freyd and David Yetter (1989). "Braided Compact Closed Categories with Applications to Low-Dimensional Topology". Advances in Mathematics 77: 156–182.
A. Joyal and R. Street, The geometry of tensor calculus, Advances in Math. 88 (1991) 55-112
André Joyal and Ross Street. "The Geometry of Tensor calculus II". Synthese Lib 259: 29–68.
André Joyal, Ross Street, Dominic Verity (1996). "Traced monoidal categories". Mathematical Proceedings of the Cambridge Philosophical Society 3: 447–468.
In fact, many things by Ross Street
Functorial Knot Theory: Categories of Tangles, Coherence, Categorical Deformations, and Topological Invariants (Series on knots & everything) by David N. Yetter (Author)
Many works by Bob Coecke and Samson Abramsky (eg. Temperley-Lieb algebra: From knot theory to logic and computation via quantum mechanics) are certainly accessible.
The Catsters youtube channel, especially the presentations on string diagrams
Functorial boxes in string diagrams by Paul-Andre Mellies Invited talk at the Computer Science Logic 2006 conference in Szeged, Hungary. Lecture Notes in Computer Science 4207, Springer Verlag.
Masahito Hasegawa, Martin Hofmann and Gordon Plotkin Finite Dimensional Vector Spaces are Complete for Traced Symmetric Monoidal Categories
Mostly these reference cover the Tensor/Monoidal Category line. I've certainly seen other work, but am less familiar with it.
