Damage logger by removing LED from circuit?
Yes you could break it off, but that can lead to broken traces. It be smarter to just cut the trace going to or from it. An exacto knife works great. Alternatively you should desolder it, or the resistor going to it.
Without a schematic, it's just an educated guess, but, no it shouldn't. I'm assuming the via on the left of the LED is Ground, and the LED only used as a simple status light.
It's possible, but unlikely. LEDs can detect light too, but there are better ways of doing it. Is the LED lit when the module is powered? Then it likely isn't being used as a sensor.