I am manipulating an $nxn$ metric where $n$ is often $> 4$, depending on the model. The $00$ component is always tau*constant, as in the Minkowski metric, but the signs on all components might be + or - , depending on the model. (I am not trying to describe physics with this metric). Can I call this metric a Minkowski metric? Or what should I call it?
If the coordinates are cover your space, are real, and there no identifications or other topology-changing shenanigans, then this is a pseudo-Euclidean space. That is the proper generalization of Minkowski space with different signatures.
If the coordinates are local, then pseudo-Riemannian. This allows curvature or different topologies, but still requires the signature (num. of positive and negative components) to be the same throughout the space. Additionally, the specific case of signature $(1,n)$ or $(n,1)$ is called Lorentzian, though having a Lorentzian metric only implies that the space/manifold is pseudo-Riemannian and not necessarily pseudo-Euclidean.
I'm confused about what that means, but it sounds like you just want coordinates in which the metric is diagonal. You could just say that, or state that the coordinate basis is orthogonal. If $\pm 1$, an orthonormal basis. It's up to you to make the $00$ component more special than that.
Just state that pseudo-something metric via reference to its signature $(p,q)$.
That link should also answer your other question, I think you could delete it or add it to this one.
The metric component (in contrast to the signature) are coordinate system dependend, so this statement is not really true. But even what you intended to say seem strage: in a homogenous space, why would the metric component want to grow with an affine paramert like that?