Shelf life of soft or medium boiled eggs?
Working in a professional kitchen where we cooked eggs in various styles. I would say from my experience: 


Hard boiled eggs will keep for 4 days refrigerated in the shell, before they become unpalatable.
Soft boiled eggs(hard whites, creamy yolks) will keep in the shell, refrigerated for 2 days.
Poached eggs, properly cooled in ice water and drained, refrigerated will keep from one morning to the next or aprox 36hrs.
Hard boiled egg whites (separated from peeled eggs, immediately cooled in icebath after boiling) can keep on parchment, drained of excess water, and wrapped in cellophane for 2.5 days. These can be used to make deviled eggs to order for example.
Hard boiled yokes, kept dry and seperately from the whites can be used to make sauces, salad dressings, deviled egg mix etc. but this only keeps 24-36 hours, in a fully refrigerated environment. So, if it's being used during a dinner service, coming in and out for each order, toss the leftovers at the end of the night.
Short-order eggs (fried, sunny side up, over easy, scrambled, omelets) made from fresh eggs, once it gets cold, forget it, it will smell nasty and taste pretty icky.
Premix egg or egg white stuff that comes in a carton is heavily pasteurized, it will keep on a buffet (hot hold) as scrambled eggs or omelets for a few hours and may still be eatable later but also icky.
Mayonnaise and Caesar dressing made with fresh raw egg yolks shouldn't be used for more than 48 hours. Even if it's kept refrigerated most of that time. A lot of restaurants make their own and I've seen several places keep it far longer than they should. But to be on the safe side, 1-2 services. So you want to make a small portion, not a huge batch.