How long can I keep egg whites (freshly separated; not processed) in the fridge without going bad?

Not very long. You can put them in the freezer if you want, that will keep them preserved for as long as you want. just thaw them in the fridge, and then on the counter when you are ready to use them.

I have had hard boiled eggs for a week and they have been fine. I would suggest, as with all perishables like milk, meat, and eggs, to place them at the coolest area of the fridge which is the bottom shelf towards the back. (Just make sure that the setting that you have your fridge on doesn't freeze items placed at the back...on any shelf.) If you have an older fridge that has the egg shelf on the door, DO NOT store eggs there...if you have noticed, appliance companies have stopped making this due to the door not being cold enough to safely store eggs. And another thing, once you cook the eggs, DON'T ever place them back in the original containers...they are contaminated and therefore should never be reused. I hope this helps. And here's something else, newer eggs are harder to peel, so boil eggs that are at least a week old (from the time of purchase!). And last but not least, if you like deviled eggs but don't have the time, make a simple egg salad and add a little DRY mustard seasoning....it tastes great.

Yes, they will be fine. The whites last longer than the yolks do. And either way, refrigerated and covered, they would last as long as the date on the egg carton. (Just the egg whites would last even longer)