Never heard of that before, room temp ketchup is disgusting. Not having it in the fridge is akin to not refrigerating cheese, it'll taste like shit and spoil quickly.

If it is not salt fermented pickles but rather those pickled in vinegar, it is good to put them in the fridge after opening the glass.

I can see it with eggs, if you don't wash eggs they're shelf stable for a couple weeks because of a natural coating. I was super confused when I went to the UK and all the eggs were on a shelf at the supermarket instead of in a cooler.