Use lowpower-libary and still use serial connection and other things
Yes, activating low power does disable the UART - but only while it's sleeping.

To use serial you just need to ensure that you only use it while it's not sleeping - and due to the interrupt and buffered nature of the Serial driver it's not as easy as just printing.

However, it's simple to achieve: just flush the serial before sleeping. This ensures that any pending serial data is sent out through the serial port before you go to sleep:

void loop() {
  Serial.println("Turning LED on");
  digitalWrite(LED_BUILTIN, HIGH);   // turn the LED on (HIGH is the voltage level)
  Serial.flush();
  LowPower.powerDown(SLEEP_8S, ADC_OFF, BOD_OFF);                      // wait for a second
  Serial.println("Turning LED off");
  digitalWrite(LED_BUILTIN, LOW);    // turn the LED off by making the voltage LOW
  Serial.flush();
  LowPower.powerDown(SLEEP_8S, ADC_OFF, BOD_OFF);                  // wait for a second
}


If that doesn't help you may need to turn serial off and on yourself either side of your powerdown:

void loop() {
  Serial.begin(115200);
  Serial.println("Turning LED on");
  digitalWrite(LED_BUILTIN, HIGH);   // turn the LED on (HIGH is the voltage level)
  Serial.flush();
  Serial.end();
  LowPower.powerDown(SLEEP_8S, ADC_OFF, BOD_OFF);                      // wait for a second
  Serial.begin(115200);
  Serial.println("Turning LED off");
  digitalWrite(LED_BUILTIN, LOW);    // turn the LED off by making the voltage LOW
  Serial.flush();
  Serial.end();
  LowPower.powerDown(SLEEP_8S, ADC_OFF, BOD_OFF);                  // wait for a second
}




Update: 

I just noticed you're using the Micro. That makes a big difference. The Micro and the Leonardo's Serial object is actually a USB CDC/ACM serial emulation, not UART.  You cannot sustain a USB connection while sleeping. So it will be killed.

To use communication while sleeping you will have to switch to a proper UART connection with a USB to TTL UART (e.g., FT232RL) adaptor to connect to the TX/RX pins and use Serial1 instead.