If she wants to keep them,  put them in the butter compartment in the fridge door andleave them there. In a few months they will be all dried out inside. If you shake one you can hear the solid yolk rattle. It could take 12 months to dehydrate properly. Alternatively, she could blow them and the use the contents for cooking.

Now that youfound the nest, she may not go back to that spot.if both eggs turned up today they would have laid one each. If you want her to keep laying in that spot,  stick a golf ball or ping pong ball there.

She would be going up the tree because chooks like to sleep high. If perches in a coup are staggered in height they will all fight over the highest one.