Mustard when opened or unopened can remain out of the refrigerator without going bad.
I've never refrigerated mustard or ketchup and just leave it out on my counter to have it ready to use because I use both mustard and ketchup often so it sits out.
The mustard should be perfectly fine out of the refrigerator but you can keep the mustard fresher by refrigerating it.
Mayonnaise however needs to remain refrigerated once you open it but mustard and ketchup doesn't require refrigeration.
But you can keep mustard and ketchup in the refrigerator to keep it fresher.