And does it similarly affect the cells of organisms of any other biological Kingdoms?
The corrosive part is, to put it simply, the hydrogen ion, not the fluoride part. Also, you are confusing fluoride (F-) with fluorine (F2).
Googling 'fluoride toxicity' doesn't readily yield information about fluorine's absorption by cells and/or reactions with cellular components such as organelles and enzymes.
I want to know how it (fluorine and/or fluoride) interacts with what to cause cell death.
The fluoride ion, not the hydrogen ion, is the source of toxicity. Fluorine (the element) is extremely electrophilic (wants one more electron pair so it can complete its p-shell and fool itself into believing that it's neon). Fluoride (the ion) has obtained only one of the two electrons it wants, and it's desperate for one more. So fluoride will steal -- or borrow -- a stray electron from *anything*, which is one of the reasons that it's so corrosive/reactive. Introduction of an electrophile to a biological system will probably disrupt electron transport (energy metabolism), bind to nitrogen-containing compounds like proteins or nucleobases, and generally -- because of its size -- prevent the molecule's proper alignment or conformation.