is there a discipline of mathematics that studies graphical versions of various operations? What I am interested about is a discipline that deals with mathematical operations that can be done graphically, in this case meaning using some kind of "structures" that are manipulated to arrive at an answer.
No number-symbols are used with these "structures" but instead the structures can, for example, contain an amount of nodes that correspond to a value. Through various operations to these nodes (such as perhaps grouping, slicing, or others) a process would then arrive to the searched for value. In the end this process would correspond to a function that can also be solved some standard way mathematically.
The reason why this would not contain any number-symbols, such as "12", but instead have e.g. 12 nodes is in the operations (e.g. grouping/slicing/etc.) that the discipline would allow on those nodes.
Is there any mathematical discipline that deals specifically with this form of mathematics? Or would this perhaps be viewed as unscientific?
 A: I'll offer a limited view of what I learned until now. Such structures do exist, take for example what JeffW89 suggested: Graph Theory. There are graph operations that do not need number symbols, there are also braids, knots and a lot of other objects in mathematics that are similar to what you're looking for. Having a trefoil knot in your hands may tempt you to unknot it, there might be some very complicated set of movementes that unknot it, but having the know in your hands alone, how are you going to do this? Having tried a lot of moves on it, how do you know that you've tried everything that's possible to unknot it?
The problem is that having these objects alone (seeing them as drawings) is extremely limited to study them. Suppose you have a huge graph and you want to find the minimum set of vertices that disconnect the graph, how can you find it? That's why we write the graph in a matrix: Because in there, it's possible to make them viable computationally (we have an algorithm to it). Take for example the example of the knot I gave you, there are ways to show that it's not possible to unknot it.
The whole power of mathematics is on converting something pictorial in some structure that allow us to prove statements about their behavior and also to compute them. So, to answer your question in a simple shot: Yes, there are a lot of such structures, but turning them into some kind of number system is what allow us (in most of the cases) to prove things about them and to compute and combine them. There's a branch in mathematics called representation theory in which they study complicated objects by using simpler objects as matrices. 
There are other cases in which the structure with number symbols is a polynomial, there are rook polynomials and generating functions (which are not really functions, and not really polynomials, they are infinite series in which one is not concerned with questions about convergence). These two representations of pictorial things help us to compute them easily. There's another classical example: Euclidean Geometry, using it's axioms enable us to build structures without the help of these number systems as an example of problem in it, take the quadrature of the circle: Basically, it's about using ruler and compass to construct a square with the same are of a circle, people tried to do it for a lot of time, but it was only proved possible by using combined properties of these figures and some auxiliary languange.

Note: I am using number system in a quite relaxed way, they require a lot of work to be well defined.

