1. The BiTalino clock speed is 8MHz so you will need to change that on the source code.
2. I never worked with the WiFi SoC ESP8266, but in order to start/stop an acquisition and choose the the acquiring frequency you need to send specific commands to the microcontroller via UART.
3. I never used Python Barebone sketch, but if you connect the FTDI to the UART pins instead of the ESP8266 you will be able communicate with the microcontroller and acquire live data directly to the PC.
4. Depends what signal you want to measure but for example if you want to measure an EMG signal the range is more or less -2mV to 2mV and the Arduino doesn't have enough resolution for that so you would need a sensor.
You can find on the following link the protocol specification for the commands and for the streamed data:http://bitalino.com/datasheets/MCU_Block_Datasheet.pdf