I get an application error message in Oracle Forms when I try to open a new window from the current one while speech recognition process is continuing.
Temporarily stopping speech recognition process using "PauseSpeechRecognition" method will solve the problem.

Can I use speech recognition facilities of DikteAPI component in other windows when I place it on one window only?
You should put DikteAPI component on every window that you want to recognize speech and call "Initialize" during the opening of that window.

Does putting DikteAPI in every window of the application use up system resources?
System resources that DikteAPI needs are allocated only for once. Therefore, adding more DkteAPI components does not increase the consumption. One thing to keep in mind here is that, "StartSpeechRecognition" method must be called once, each time the current user is changed.