It is recommended to use a local testing server for a number of reasons, the two biggest ones are:
1) Many remote hosts will restrict access to the MySQL Server by outside connections. meaning that the MySQL server will not allow your computer to connect to it, only the hosts server can connect. Some hosts will offer a way to add your IP address to the allowed list to get around this, but some dont.
2) Even if you can get a connection to the remote MySQL Server going in Dreamweaver, it will frequently get dropped by Dreamweaver and cause the extensions to hang on trying to reestablish the connection.
Setting up the local testing may seam like a lot of work at first, but it greatly improves the performance of Dreamweaver to connect to a local database over a remote database.
The error message you report in the second message means that the Web URL you specified in the server setting cannot be connected to.