If the link is said to be for an ArcGIS REST connection, that's the web server you use.
Choose File - Create New Data source.
Choose Web Server: arcgisrest for the Type.
Enter the connection string in the Source box. Click the ... button so in that next dialog you can press the Test button to see if the connection string works. If it cannot make a connection in that dialog, it's not going to work otherwise.
You don't say what the first link is supposed to be. Trying it in a browser gets a form (you can always try a connection link in a browser). I see King County has some sort of geo-fence in action, so their servers don't allow connections from foreign countries (I'm in Europe). I can connect using a browser that runs through a VPN that lands in the US, but not from France, so I can't help you with that.
The last two links when you try them in a browser report errors. If a browser can't connect, Manifold won't either.
To learn how to work with various connections, see theExample: Style Applied to an Image Server Image topic Try connecting to that server using step by step instructions in that topic.
The CSV Servers topic shows some more examples.
Get the ArcGIS_REST_Servers.map collection of web servers from the Product Downloads page, so you can see many examples of those.
The usual issues with servers is that they're not turned on, they want a login, they don't honor the standard (json is notoriously flaky and not geojson in many cases) and so on. And some just fall into the gray area of weird configurations, just another variation on loose quasi-standards.