How can I save/load my Entity from a different database in Drupal 8?
At least if the "external" database is on the same server, there is an easy solution. The base_table of an entity can refer to a different database using . as a separator.

Just adapt the EntityType annotation in the comment above your Entity class. The following example uses the table myentity_table from the my_database database for this Entity's data:

@ContentEntityType(
 *   id = "myentity",
 *   base_table = "my_database.myentity_table",
 *   [...]
 * )