The beeper is going.
1. How about 'the telephone is going or ringing'? which one is correct, 'going' or 'ringing'?
2. Is it correct to say, 'The door bell is ringing'?
3. How about 'The alarm clock is ringing'?
1. The telephone is ringing.
3. Yes. You could also say, "The alarm clock is going off." That doesn't mean that it is turned off, it means that it is making noise. Your first sentence should probably be "The beeper is going off."