If I fail over one database, do the others that share the same mirror endpoint fail over as well?
@mrdenny's answer is accurate that failing over one database will not result in all the other databases failing over as well. 

However just to give more overview of what a database mirroring Endpoint is:

From BOL,

Connection management in Microsoft SQL Server 2005 and later versions is based on endpoints. An endpoint is a SQL Server object that enables SQL Server to communicate over the network. For database mirroring, a server instance requires its own, dedicated database mirroring endpoint. All mirroring connections on a server instance use a single database mirroring endpoint. This endpoint is a special-purpose endpoint used exclusively to receive database mirroring connections from other server instances.

To be more flexible, you can use T-sql for doing manual failover:

Turn mirroring off (break mirroring) on the mirror DB.

ALTER DATABASE &lt;&lt;DB NAME&gt;&gt; SET PARTNER OFF


Set the recover mod the DB to RECOVERY

RESTORE DATABASE &lt;&lt;DB NAME&gt;&gt; WITH RECOVERY


Fix and orphaned users if you are using SQL authentication

EXEC sp_change_users_login ‘Auto_Fix’ , ‘&lt;&lt;username&gt;&gt;’


EDIT:

Since you are using high safety mode (as per your screenshot), you can use the following from the principal server

ALTER DATABASE &lt;&lt;DB NAME&gt;&gt; SET PARTNER FAILOVER


Doing so, only the role reversal happens and mirroring direction is reversed - principal becomes mirror and vice-versa. Note that MIRRORING is not BROKEN.

Note that per this MSDN article


  Manual failover can be initiated only from the principal server.


If you want to automate the entire failover then refer Database Mirroring Automation