When I change in line 117 LOW->RISING, not even the name of the code (SparkFun ADXL345 Accelerometer Hook Up Guide Example) is displayed entirely
That's because serial output is written into a buffer and effectively sent later on in interrupts. Insert a Serial.flush() after line 51 in your code and the complete line will display before the code is going on.
After out commenting lines 121 and 124 at least it works somehow. The name is displayed, and the message "Going to sleep" pops up.
That is only display because of the delay() calls after it.
BUT every 10 seconds it displays the current acceleration and it says "going to sleep" again.
That feels like the command
LowPower.powerDown(SLEEP_FOREVER, ADC_OFF, BOD_OFF);
is not a sleep_forever command:=)
Is it exactly 10 seconds? Did you check the ADXL345 interrupt line?
The SLEEP_FOREVER just means that the watchdog timer is not activated. Any other wake up source is still active.