My recommendation is to download only the part needed for editing. Downloading and changing are much faster and the risk to change something by accident is much lower.
JOSM is great for editing but very limited for viewing a whole country because all details get drawn on each zoom level. To circumvent this different filters at different zoom levels could be used. But there are much better programs for viewing a map which are fast and show more details at higher zoom. Some offer even customizable rendering and routing: OSM wiki: Software