[deep thinker]

Hi Nick, heres answers according to what I do and experience with mine as well as what I have seen from other divers I dive with.
How I should take care of my freediving fins- Just like the rest of your gear, a good rinse and clean after every dive and store out of direct sun in a cool dry place. Check also for rubber fatigue and stress marks every now and then to ensure best service. If theyre carbon blades check for small cracks, these I think can be drilled out at the end of the crack and superglued as a temporary solution but Id replace the blade asap.
How should they be stored, transported ect- Store them flat in the bottom of a cupboard or on a shelf or Ive also seen guys drill a small hole in the front of the blade and hang it off a cup hook. For transportation I put mine in the top of my diving bag, just make sure theres no heavy objects (weightbelts) or sharps (knives and speartips) that can damage them.
What could damage them in and out of the water- In the water if you over power them they can get stress marks, in plastic fins white marks looking as if the plastic is stretched and it mite also make a permanent bend there, cracks in carbon fins. Scratches and bends or once again cracks in carbons when banged on rocks in surf or while hole and reef hunting, scratches is gonna happen though so I wouldnt stress too much about that. Out of the water basically the same, dont throw them around and dont put or throw heavy objects onto them.

Sensible care when working and looking after any of your gear will ensure longer use and possibly better performace.