Almost. An mxd is a project file that contains (amongst others) layer definitions. You might say that an mxd is a collection of layer files, but besides that, it has for instance a pagelayout, a map, and other stuff. A layer in an mxd can be exported to a lyr file, and a lyr file can also be imported. However, the mxd does not hold references to lyr files.
Mxd or lyr files do not contain the actual data, but only hold references to it. Data can be stored in several formats, the shapefile probably being the most well-known. Other formats are a file geodatabase, an enterprise geodatabase (Oracle, SQL Server, ...), GeoTiff, etc. See the documentation for more information if you like.
When you publish a map to ArcGIS Server an mxd will be converted to a format that is more suitable for ArcGIS Server. This format does not support everything that you can store in an mxd (e.g. not all symbology can be converted), but basically, it is as you say: One mxd, one service. Once a service has been published, there is no longer a link to the original mxd, although the filename is stored somewhere in the metadata of the service.
BTW The data may need to be copied to ArcGIS Server as well. This is part of the publishing process and is taken care of automatically. Only if the data is already accessible by ArcGIS server, either because it's on a file share, or in an Enterprise Geodatabase, copying is not necessary.