Is the condition number specific to a particular matrix A and nothing more? Given a particular matrix A, is its condition number fixed, or does it depend on the context of discussion? For example, would talking about it in the context of Algorithm X or Algorithm Y change the condition number because the components of the procedures are different?

The condition number of an invertible square matrix $A$ over the real numbers is defined as the norm of $A$ times the norm of $A^{-1}$. Notice that this definition does not specify which norm we should use. Hence given a matrix $A$ we have as many condition numbers as matrix norms. However, the vector space of square matrices over the reals is finite-dimensional and norms over finite-dimensional vector spaces are equivalent, in a certain sense. Consequently, the various condition numbers of a given matrix are equivalent in the same sense. Note that the condition number is a relative measure of the numerical behavior of a matrix. Now, the condition number is a generic notion, it does not depend on any given algorithm, e.g. an algorithm for matrix triangularization. Conversely, the dependence or non-dependence of an algorithm on the condition number of the matrix, determines the quality of the algorithm. For example, the accuracy of Gaussian elimination depends on the numerical properties of the matrix, while the accuracy of the $QR$ factorization does not. To use your phrasing, the different "components" of different algorithms might determine whether the algorithm's accuracy depends on the numerical properties (condition number) of the matrix or not.
