Although this response is a bit late, perhaps this perspective may help nonetheless. It only addresses the question about functoriality of the GNS construction.

The GNS construction is not quite a functor, though as Nik Weaver's response indicates, it has a certain functoriality associated with it. The GNS construction, is, however, (*almost*) a natural transformation that satisfies a universal property, which is indicated in Westerbaan's response. Let me make this more precise.

**Background**: There are two relevant functors on the category of $C^*$-algebras. One is the functor that associates the set of all states to every $C^*$-algebra and the function that pulls back states to states to every $C^*$-algebra $*$-homomorphism. This is the ``states'' functor $\mathcal{S}:\mathbf{C}^*\text{-}\mathbf{Alg}^{\mathrm{op}}\to\mathbf{Set}.$ There is another functor that sends a $C^*$-algebra to its category of representations and a $*$-homomorphism to a functor between categories of representations $\mathbf{Rep}:\mathbf{C}^*\text{-}\mathbf{Alg}^{\mathrm{op}}\to\mathbf{Cat}.$ A representation is not enough to produce a state, so we add the additional data in the category of representations to include a normalized vector. Morphisms in this category are then taken to be intertwining isometries preserving unit vectors. Thus, we write $\mathbf{Rep}^{\bullet}:\mathbf{C}^*\text{-}\mathbf{Alg}^{\mathrm{op}}\to\mathbf{Cat}$ for this new functor. The codomains of $\mathcal{S}$ and $\mathbf{Rep}^{\bullet}$ are not the same so it does not make sense to compare them. By viewing every set as a discrete category, we can take the codomain of $\mathcal{S}$ to be $\mathbf{Cat}$ as well. To distinguish $\mathcal{S}$ from this functor, let's call the latter $\mathbf{States}.$

**Statement**: The GNS construction furnishes a *pseudo*-natural transformation $\mathbf{GNS}^{\bullet}:\mathbf{States}\Rightarrow\mathbf{Rep}^{\bullet}.$

It is *not* a strict natural transformation. Technically, it is a 1-morphism in the 2-category of functors from $\mathbf{C}^*\text{-}\mathbf{Alg}^{\mathrm{op}}$ to $\mathbf{Cat}.$ The objects here are functors, 1-morphisms are pseudo-natural transformations (natural transformations that satisfy naturality up to a higher coherence), and 2-morphisms are modifications.

**Universal property**: The GNS construction is actually part of an adjunction in this 2-category mentioned in the previous paragraph. It is left adjoint to the natural transformation (which is an honest natural transformation) that takes a representation with a unit vector and simply pulls back the canonical state associated to this unit vector to the underlying $C^*$-algebra.

[There is supposed to be an image here---sorry, not enough reputation points---please see the introduction in the paper below]

More details can be found in https://arxiv.org/abs/1609.08975 "From observables and states to Hilbert space and back: a 2-categorical adjunction," which was coincidentally posted not long after your question, though I was unaware of this until now.