You don't state how old your chickens are. Leave oyster shells in a container in their coop (Like next to their entrance/exit door) they will eat it as they need it. Make sure to check and refill it when getting low. Mine eat the chunk oyster shells and I'm not comfortable with giving them...
Shadrach is correct. Outside of existing fencing. Keeps predators from digging underneath existing fence. Do you have the run covered? That should be a concern as well.
No, we don't trash all eggs. Eggs can be cold but as long as the shell is not cracked we keep them. If shell cracked, they now have bacteria that could have entered through the crack; we will toss them, feed them back to flock, or to dog, etc. Happy Chickening