How is the interaction between two strings (in string theory) computed or quantified? What properties or quantities of two strings are considered when computing the interaction between them?
Is there a conventional way to represent interaction of two strings diagramatically? Or even better in form of an equation that takes a defined number of strings as input?
 A: Conventionally, one approach in normal quantum field theory is to compute correlation functions, and through the LSZ redution formula we know those are related to scattering amplitudes.
This approach fails when the gauge symmetry is a diffeomorphism, and so there exist no local off-shell gauge invariant observables to work with.
To remedy this, we can take the points we would normally consider when evaluating a correlation function to infinity, and each leg of the diagram corresponds to a free string state with some momentum.
However, we also have conformal transformations at our disposal, to bring points at infinity to somewhere of finite distance. In addition, thanks to the state-operator correspondence in conformal field theory, we can relate any state to a particular operator, known as a vertex operator in this context, which we insert on the worldsheet.
Just as in QFT we sum over all possible diagrams, which means over all loops if we choose to organise them that way, we sum over Riemann surfaces of all possible topologies.
Since in string theory we are evaluating - in place of a Feynman diagram - a Riemann surface with operators inserted on it, the "loops" become holes instead on this surface.
Without going into the technical details, it turns out at each loop or "hole" order, you only need to compute one diagram. As you can imagine then, the first to consider is a sphere, followed by a torus, and so on.
I will conclude by saying it is possible to define the five string theories we know based on this formalism. In other words, we do not need to specify a Lagrangian plus certain choices, we can fully specify the theory by saying how we compute these Riemann surface diagrams.
A: An important clarification is needed. The "miracle" of string theory is the discovery that a theory of strings can be consistently quantized. After quantization strings have a discrete (but infinite) spectrum of states that possibly depend on continuous parameters (such as position or momentum), exactly the behaviour expected for any reasonable quantum system.
The properties taken in account when two quantum strings scatter are the transitions bewteen initial and final states as in any ortodox quantum mechanical scattering problem.
String theory is extremely respectful of the rules of quantum mechanics and quantum field theory. The difference rely in how string theory use quantum mechanics to predict new physical phenomena. See  Why string theory is quantum mechanics on steroids.
Suggested readings:
All of string theory's power, beauty depends on quantum mechanics
First-quantized formulation of string theory is healthy
First stringy steps: how a young fieldist expands her mind to become a string theorist
String theory textbooks
