OSM is the data!

The various slippy map tile renderers use that data, although they may use third party data (e.g. ground heights) as well.

Generally they will load the data into a database before rendering, but they are free to do it anyway they like.

If a way is missing a name, it is missing a name, and the renderer will not render a name for it. If it is part of a named road, that is a mapping error, that the renderer should not fix.

You put ways together by their shared node numbers.