Does the geometry affect the way an opaque material looks, or does only the UV mapping matter? I have a complex poorly built model with all triangular faces, no edge loops, unconnected vertices, etc. and was wondering if the geometry affects how the material looks? It is a metallic material.
The triangle faces do not matter when adding materials, the triangles inhibit your use of certain tools and modifiers, and the ability to easily change the model.
The model will be triangulated by the render engine any way, not actually adding edges, but under the hood quads are calculated as two triangles.
The bottom model is way better and much more efficient. I'm guessing this has a subsurface modifier of at least 3 subdivisions.
The top model doesn't need holding edges, the mesh is dense enough to shape the edges of the model. Holding edges are used to help shape the model when using a subsurface modifier.
As for the unconnected vertices, It is not possible to see which vertices are unconnected from the image.
At a guess I think this maybe done to separate parts of the model into islands for texturing. We would usually keep vertices connected, removing doubles for example, to improve performance and efficiency. It is much better to achieve the same thing by marking seams and UV unwrapping the model.
To answer your question, if the triangles are on the same plane, meaning a flat surface divided up into triangles, then you would be ok.
However, in the case of this model, the triangles on the curved ends, most definitely it will affect the material. With a metallic material you will clearly see the distortion of the reflections at every edge of each triangle.
My personal preference when modeling anything, is to aim for a model low poly enough to be efficient, but high enough to use a maximum of two subdivisions on a subsurf modifier.
