How did they start sending messages to other's phone?
You just send the message from one phone/pager to another one. All you need is the microwave running and a running CRT (Cathode Ray Tube), plus the phone number of the receiver. You configure the microwave, turn it on and wait for the message to be sent.

Earlier in the story, they needed to connect the phone with the microwave, to determine, which message should be affected by time-travel. 


   However, Daru wrote a program, with which you could set the number and various different things, so it was possible to send these D-Mails without the need of connecting the phone to the microwave.


Regarding your time-question: Daru applied many modifications to the microwave, so you could set any running-time. The microwave won't run the whole 24 hours, it just needs to be set to that time.