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You can use any WiFi module: ESP8266, CC3000, HDG204, etc. Even some bluetooth modules have the required bandwidth (64kbps). In this last case, you will need some sort of dongle that shows up as a serial port in Windows so you can connect the bluetooth module to your PC and BitVoicer Server.
You do not have to open the sketch (Arduino IDE), but BitVoicer must be open and running. If you use BitVoicer Server, you do not have to open anything.
To enable communication with your Arduino, yes.
If you do not have a license, BitVoicer will be unable to open the serial port. However, I do not think it would lead to a Speech Rejected response. I suggest you contact BitSophia for that.