Find out what entity type an entity is? How can I find out what entity type an entity is, when all I know is the entity ID?

Or is there a way I could list all the entity types on my website (or even via database)?
You can't.

Different entities use the same ID. 

E.g. you can have a user with ID 42, and a node with ID 42, and a taxonomy term with ID 42. So just knowing the ID is 42 is not enough to know what type of entity the ID belongs to.

This is why entity_load takes both the entity type and the ID as arguments.

To list all entities use entity_get_info().