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The sticky substance you may get on your fingers while disbudding is a sugar substance (like a nectar) that the plant exudes during this stage presumably to help the petals to slip on each other as they are putting on tremendous growth during the month. (I usually come in from disbudding with sticky thighs on my jeans, but at least it is washable, and just tastes like sugar if you lick your fingers.) This sugar substance may attract ants which like to feed on it. Ants are not a problem to the plant unless they make a nest at the base.

Generally fertilizer or compost can be added to bottom of a planting hole, taking care to cover that will soil. You do not want the roots to touch the compost.
Direct contact can burn the plants.

You can then reapply a light fertilizer when the plants are actively growing in the spring.

To avoid fertilizer burn you should make sure that the fertilizer does not come in contact with the Peony crown or within 6 inches of leaves or stems.
After applying fertilizer you should water to encourage the fertilizer to mix with the soil.
To know accurately what fertilizer is needed, you should have a soil test. A balanced 5-5-5 or 10-10-10 is a good choice. Avoid high nitrogen as this will leave you with lots of leaves and no flowers.

Water out the plants to try to move the fertilizer away from the plants.

Depending on how severe the damage is, they may recover over some time.

https://www.gardeningknowhow.com/ornamental/flowers/peony/peony-care.htm

Pruning should take place after your plants have finished flowering.

Here are some articles that will help you.

https://www.gardeningknowhow.com/ornamental/flowers/peony/peony-pruning.htm

https://www.gardeningknowhow.com/ornamental/flowers/peony/dividing-peony-plants.htm

This link will help you with more information.

You can look for cultivators that are resistant to Powdery Mildew.

I have this issue with my Bee Balm in my garden and though I could tear them out and replace, I have had luck with spraying them with Neem Oil every 2 weeks from the time they emerge from the soil.

https://www.gardeningknowhow.com/ornamental/flowers/peony/powdery-mildew-peonies.htm

The foliage of Itoh peonies does look just like the foliage of herbaceous peonies. However, the plant gets larger and the flowers get larger. This article should help
https://www.gardeningknowhow.com/ornamental/flowers/peony/growing-hybrid-itoh-peonies.htm

No, you will want to let the plant die down naturally, and then cut the stems down to about 3 to 4 inches above the soil in the fall.
This article will help you with this task.

https://www.gardeningknowhow.com/ornamental/flowers/peony/peony-pruning.htm