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Early fall or summer is the best time to move them. You want cool, wetter weather and these seasons provide that.

Yes, that should be fine. As long as the container plants are sheltered from the cold, they should overwinter in either place. While the lily plants will not require any care, as watering could rot the bulbs, the hydrangea may need to be watered once a month to keep it from drying out completely.

White lilies prefer at least six hours of full sun and well-draining soil. They should be planted in the early spring, after the last frost of the season.

It's difficult to pinpoint the exact pest; however, treating the plant with neem oil is usually effective on many types of pests. Here is more information: https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm

The only thing that comes to mind may be some type of lily variety. Could you provide a description of the plant or send us a picture? This would help us to better identify it for you.

Technically, you can transplant at any time during the warm months, but when temps are very high, the water stress on the plant greatly increases the chances that the plant will die. If you transplant during high heat, provide lots of water for at least a week or two after transplanting. Providing temporary shade for the plant is helpful as well.