Hmmm interesting one...
With regard to changing the cable - Beocoms/Beotalks don't like "standard" cables - I figured this out after lots of headscratching. They use a different wiring, and that may be your problem, or at least that part of it whereby you don't hear anything on the speaker, and the phone beeps are not recognised. But....
...this doesn't explain why the dialtone stops when you press some keys, but not others. Also, if the dialtone stops, when you press some numbers (ie: 1) then pressing 1 multiple times should kick up a "number not recognised" voice message from the exchange.
What's the origin of your phone? Do you know if it is an Australian issue, or has it been imported there? it could be it is making the wrong type of beeps... I know not all countries have the same DTMF frequencies, but not any details, so if anyone else can help......?
Failing something like this, it sounds like you may have a faulty Beocom, but I'd certainly look at the cable first, as my 2500 will not work at all with a "standard" cable...