1. Don't give much fertilizer if any during a drought period. If you have rains then feed. Many people never feed their peonies. I do it in the middle of winter. One of those I can't stand being in the house moments.
2. Go ahead and deadhead. Cut the stems just above the stem leaf.
3. No he can't mow down the foliage. Let him start this and he will mow down something else without asking. Experience was a great teacher:) Tell him just as he needed food, sunlight and rest to become what he is today. So do plants. This said if he does it anyway peonies have survived for many years in places they have been mowed every year. If the plant is new it is best to leave the leaves.

Peonies without bloom are quite an attractive shrub and what is to be gained by mowing them down? Properly prepared soil does not need fertilizer, but if you must fertilize go easy on the Nitrogen. Leaving dead flowers on peonies is ugly and serves no purpose. Al