And does it matter what system your data layer begins with if you choose the right system for your project? Do some applications use more precise projection systems than others?
From my admittedly incomplete understanding of projections, they should all be equally precise, at least within reasonable limits. Its just a matter of using more decimal places. The main difference between projections are what aspects are "true to life" and which aren't. For example, some projections preserve N-S distance, others E-W, others preserve area at the expense of distance, still others preserve constant bearings or true bearings, and some just try to minimize a variety of errors. If you're working in a small area, I'd recommend using the appropriate UTM zone for your area, as the distances are "true" N-S and the zones are set up to minimize E-W distortion within a zone. If you're working on a larger project, use UTM if it is oriented N-S, or a Lambert Conformal Conic if it is oriented E-W. For a fuller discussion of what you need to know, read the answers to this question.