I am endeavouring to merge some faces within a rectangle so that I can push/pull a rectangle up without having to hold shift and select multiple surfaces.
One can only guess without working with the model firsthand.
Sharing the model would definitely help us help you.
My guess is you’re experimenting with SketchUp’s Layer system before understanding how SketchUp’s
geometry behaves, whether it’s visible or not.
What do you mean by merging faces? If they are within the same geometrical group and are coplanar, SU normally does that for you. You do get Z fighting with coplanar faces when you have different geometrical groups. And of course faces will not merge if they are not coplanar, which may be your issue here. As @Geo has said, no one can really say without seeing the file.
Hiding things randomly and/or having some edges/faces NOT assigned to Layer0, is a well documented recipe for a terrible headache !
Also please use View > Hidden Geometry > ON and re-snap the screenshots.
Also Viewing in Monochrome mode [preferably using a Style which has distinctive front/back default colors] can help avoid the confusion of the multiple textured materials…
I suspect that we are looking at a multi-faceted surface, perhaps with some non-coplanar or reversed faces ?