Commutative diagrams are, to put it simply, a (very) handy way of writing systems of equations in categories. Instead of having rows of symbols and do symbol-pushing you reason diagramatically and geometrically.
Now, about your questions, I am not sure I understand all of them so my answer may be way off, but here it goes anyway.
Which "background assumptions" are needed for a commutative diagram to make specific sense?
Without going (too) technical here, diagrams in categories are oriented graphs where the vertices are labeled with objects and the edges with arrows and to every path we associate the composite arrow. The problem is that the composite is not well defined in the presence of loops, e.g. think of a single vertex with two loops labeled f and g: does it denote the composite fg or gf? If your labeled graph does not have loops then you are all set.
Note: making this all formal and rigorous is something of a chore and for a little gain. One possible way is to define diagrams of shape $G$ (a graph, for example a square) in a category $A$ as functors $FG\to A$ where $FG$ is the free category on the graph. Then a commutative diagram of a shape $G$ is a functor $FG/R \to A$ where $FG/R$ denotes the quotient of $FG$ by a congruence $R$ that "forces" commutativity, that is, the equality of certain (all) parallel pairs of paths. You can then define the gluing of diagrams along a common diagram as a certain pushout, which allows you to prove that if the smaller pieces of a diagram are commutative then the whole thingy is also commutative, etc.
Which conventions are used? (e.g. for existence and uniqueness of arrows)
I am not sure understand what you are trying to ask here. What problem of existence and uniqueness are you referring to? The existence and uniqueness of a composite associated to a path was treated in the previous answer.
When is it important to label the objects and arrows?
Unless there is no ambiguity about what arrows and objects you are mentioning (or you simply do not need them), feel free to not label the graph. In my experience, this happens very rarely so I tend to label just about everything.
When are verbal explanations necessary?
No general rule here. The best that I can offer is use common sense and know your target audience.
Does every conceivable diagram mean something?
No, see my first answer.
Are there (graphical) analogues of commutative diagrams for other (relational or algebraic) structures than categories?
Yes, there is a fairly sophisticated graphical calculus for braided monoidal categories with duals involving tangles. There are also higher dimensional generalizations of diagrams to n-categories but here the problems of assigning a consistent composite to the corresponding notion of graph are much harder and their computational usefulness is rather limited for reasons that should be obvious. Even 2-diagrams in 2-categories are already very cumbersome. This is actually one of the reasons why higher dimensional category is inherently harder: you do not have a handy graphical calculus and computing anything can range from a pain in the butt to a pain in the (insert ultra-sensitive bodily region here).
Hope it helps, regards.