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.
Half right. Warm ketchup tastes better but it does spoil faster. It's ok for a bit thanks to the salt and vinegar, but definitely not the life of the bottle.
Generally, it's better. Things like mustard and most hot sauces are acidic enough to hang out outside the fridge. Ketchup is in less of a good spot this way and more prone to mold or spoilage.
Not sure about ketchup in other countries but American ketchup usually has a surprising amount of sugar in it, which is obviously not great for shelf life outside of a fridge.