Most will have foliage all summer. The leaves are how they make the food to provide flowers next year. You may interplant with about anything. The only thing you need to worry about is acidity and if the roots are too close to the peonies.

After blooming for a neater look cut the flower stems down to below the leaves. Watch your plants some will provide yellow to bronze to red autumn foliage if autumn cooling is long enough before freeze stops the process. You might want to mark which ones you like the best in case you wish to redo your beds.

You may plant anything that wouldn't need a frequent, and better not at all, divisions, as Peony don't like their roots to be disturbed.
For that reason bulbs (daffodils for before and lily for after the peony bloom) are not suitable.
Depends on a layout of the bed you may consider to plant group 3 clematis and let it grow without support i.e. meandering 'free style' in a bed.

Sorry, I'm a let nature take its course. I don't divide my bulbs. My answer was based on your statement that the peonies are in clumps. you would not plant right next to the peonies at least a foot or more away. There was a speaker several years ago that advocated planting lily's close enough to the peony but far enough from the roots that the lilys pushed through the foliage of the peony which supported the lily and gave additional flower time to the foliage. Most annuals without an extensive root system could be planted in and around the peonies. There are also books on companion planting and cottage gardening which you can get some ideas from. If 2/3 of the bed is free area experiment.

After the blooms fall off, the "seed looking' mass that is left, are they seeds that can be replanted? They look like seeds....