In the days before refrigeration, food was preserved by salting, dehydrating, and pickling with vinegar.
Mustard is pickled with vinegar so it could stay outside the fridge. Other pickled vegetables could be too but I think they taste better cold.
Ketchup/catsup is a toss up to me. I think it taste best at room temp, but I don't know how well it keeps at room temp.
Mayonnaise is emulsified egg and oil with spices, so I think it would be liable to spoil quickly
Next time you make a grilled sandwich, try coating the outside of the bread with mayo. You may like the difference.