The problem is with your setup. Now that I have seen your site definition I can diagnose the problem and give a bit of an explanation of DW's expectactions.
First I have to criticize the use of FTP. By sending your data over FTP, you basically send your username and password in plain text all the time. It's not DW's fault, but rather the fault of the protocol. I'd suggest having a long discussion with your web host about using SFTP, which utilizes SSH, or FTPS which uses SSL to encrypt your connection.
With that said, that leads to our first problem the Web URL. Now when setting up DW, I will admit it is a fault of DW that it adds the Root URL to the Domain. However, you will want to remove the "/public_html" from the end of the Web URL address. The Root URL simply says when you connect to FTP, connect to this folder. The Web URL is what the browser sees and while the browser is looking in the public_html folder, it is transparent because it is something processed by the server. So only your domain or IP is needed. You do not have to use your IP here but can if you choose to do so.
Next, since you have confirmed a successful connection with the Workbench, I will assume your IP is correct on the server for the remote access and has not been changed by your ISP.
Now is the tricky part and goes back to your original post about why DW adds your domain to the address. The way DW connects to your database is not the same as the Workbench. What DW actually does is connects you to the server as if you are on the server, thus the address in your SQL Connection window will actually be the same as how you configure your Joomla script, that being "localhost". From what I have read from Acehost, I am not under the impression that your mySQL database is on a separate server from your web server. It's the old conundrum of what happens when you dial your own phone number. The answer is you get a busy signal.
So if you make the change to the Web URL, and then change the mySQL address to "localhost", from everything I see you should be able to make a good connection to the server. You should leave your Joomla configuration file as "localhost" as you are now testing with the live server.
Now just to add one more comment on this. Typically DW is setup so that your testing server is not your live server so that you can make these changes without having to upload everything to your live server. So as a best practice it is typically recommended to bring a copy of your database down to a local MAMP/WAMP server and then push the changes to the Remote Server once you have completed your changes. Also to prevent always overwriting your files in this scenario you may want to go to the DW preferences -> Preview in Browser -> and enable Preview Using Temporary Files otherwise when you go to preview if you made any changes to your core files DW will save, upload and overwrite automatically for testing.
If you run into any problems implementing this please post back.
