yes they should be wrapped in cling wrap and stored in fridge until eaten
Once it's open, any mayo needs refrigeration.
No. Vegemite will store without refrigeration.
All pickles need refrigeration, once the jar has been opened.
The lifespan depends on the environment, and the weather. A pumpkin lasts longer if it's in a cool, dry, shaded setting. Cool storage such as refrigeration will add to its lifespan. In the case of carved pumpkins or pumpkins on display, they can be sprayed with preservatives or painted with sealers. Generally, from my experience, pumpkins last about three to five days. Pumpkins are annuals so the answer is a year. The fruit of course can be kept longer.
if they are broken open they will eat the insides
no,
Yes it does need regrigeration.
Refrigeration
It slows their spoilage.
No
No
In my experience, a tuna fish sandwich can survive for some hours without refrigeration - you can take one to school with you in your lunchbox - but for any longer period of time, yes, it would need refrigeration.