Remove the dialer switch-box and connect your broadband router directly to the main phone point -- does the DSL connection come up then?
If so, it sounds like removing the dialer switch-box breaks some or all of the extension connections. You may need to rewire the extensions to ensure they are connected to the phone line properly.
I've never seen or used a dialer switchbox so I'm speculating somewhat above. I am very familiar with how ADSL should work though. I'm also assuming that (since the router connects sometimes) you have the DSL filters connected correctly (it's easy to accidentally connect the DSL router to a port that has the DSL frequencies filtered out!).
As for router diagnostics, usually you can access a status page by visiting the router's IP address in your web browser, e.g. http://192.168.1.1 . You may need a password to access it though. In this case a status page is unlikely to help, as both LED indicators being off is pretty definitive.