I'll give my opinion and some of my reasoning to try to help.
First, I assume you are in Massachusetts by your name. I found that it sometimes helps people answer questions if they know what climate you are in. You can go into profile and set it up to show your location. It took me a little while to start doing this, but I finally did.
In theory and depending on your climate, yes, the eggs will stay safe to eat as long as they don't freeze or get too hot. Many people have posted on here that they store an egg at room temperature for a couple of weeks and it still is fresh enough to hatch. A hen will normally lay a nestful of eggs before she goes broody. At one a day, this will be 10 to 12 days. If they are able to hatch, they should be safe to eat.
I do not think it is a good idea. First, leaving eggs out like that is an added attraction to predators, especially at night. I'm thinking especially rats, but others could be atttracted. And once they find a good reliable food source, they will be back.
Leaving eggs in an active nest like that greatly increases the risk of one getting broken by accident. Then the other eggs get coated with raw egg. Egg shells are porous. They have to be so the developing baby can breath. So an egg with rotten egg on the surface will take in bacteria and also spoil.
An even bigger problem is that, if an egg, gets broken, the hens will eat the shell and all the egg they can. This can teach them to be egg-eaters, which means you get no eggs at all and have a real mess in the nest.
I don't understand your circumstances as to why you can only collect the eggs once a week. That raises a lot of questions about how you are going to see that they are fed, watered, an protected from predators, but that is not your question. I think you will be a lot better off giving a neighbor some of the eggs they collect if you can arrange that.
