Mermaids cannot die of dehydration, but I think this is by design. They just get stinky and dried out. Plant sims don't die of dehydration either, so it's consistent.
Mermaids cannot die of hunger despite the starve to death in x hours expiring. Nothing happens. I think this is a bug.
Mermaids can die in a fire and be electrocuted. Haven't tried other deaths, but I assume they mostly work. If pufferfish doesn't work, then that also seems intentional to me.
Now about dying from old age. This might be working as intended. Mermaids have an extended life span. It's a very long time before they die of old age. All of their extra time is allocated to the elder stage as per the sims logic. I have MC Command Center installed and it tells me they have 210 days for a mermaid on normal lifespan who just became an elder, I think. I manually set her lifespan to the max days in lifespan for elder, 47. It still said she had 164 days left. I had her eat 2 mermadic kelp to turn her back into a human. She immediately died of elderly demise. So...work around...once your mermaid's age bar is glowing as an elder, have them eat 2 mermadic kelp if you're ready for them to go. Kind of sucks for ones in the world, but you can ask them to join your household and then have them do that.