[tordanik]

Exports for countries and smaller regions (in the .osm.pbf Protobuf format as well as some others) are typically provided by third party services such as Geofabrik's download server: https://download.geofabrik.de/ It's linked from the "export" tab on osm.org, but could probably be made more visible.

As far as APIs for OSM raw data go, Overpass API is the most popular open source option. The osm.org API is neither suitable nor intended for building non-editing apps.

When it comes to keeping a rendering database current, of course, most people use the minutely diffs mentioned by rmc.