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Moth Balls is not a recommended product to use in the garden for the safety of humans and animals.

Here is an article that has more information.
https://www.gardeningknowhow.com/plant-problems/pests/pesticides/mothballs-in-gardens.htm
https://www.gardeningknowhow.com/plant-problems/pests/animals/cat-repellent.htm

Depending on the variety Honeysuckle berries are non-toxic to mildly toxic.
Cutting the berries off shouldn't effect flowering, as the birds usually eat the berries.

The photo is a bit fuzzy, but it could be black spot (Diplocarpon rosae), a disease that is most known for attacking roses but also attacks honeysuckle and several other plants. This article has more information:

https://www.gardeningknowhow.com/plant-problems/disease/black-spot-fungus.htm

You will need to do a little work prior to moving the honeysuckle vine.
Cut the plant back by 1/3 2 months prior to moving the plant. This helps the roots reduce the plant they need to feed.
At the same time cut down about 10 inches into the soil all around the base of the plant in about a 12" radius.
Use a sharp shovel but do not lift the plant. This will encourage the plant to grow out new short roots that will help feed the plant after transplanting.
This time frame should be perfect for a fall move for the plant.
Dig up the root ball from the existing cuts you made. Move the plant immediately to it's new planting site.
Use the link below to refresh you on the growing conditions. Make sure you plant the vine at the same level it was previous and water well.

https://www.gardeningknowhow.com/ornamental/vines/honeysuckle/honeysuckle-vine-care.htm

Honeysuckle prefer a rich, humus soil that is well draining.
Here is a link with care information for this vine.

https://www.gardeningknowhow.com/ornamental/vines/honeysuckle/honeysuckle-vine-care.htm

Yes, you can plant out the Honeysuckle now. Keep the plant watered up until the ground freezes to make sure the roots become established.

You can add 2 to 3 inches of hardwood mulch to help with moisture retention.

If you choose to winter it over in a pot, just keep it in a sheltered area. A garage works well for this. Check the soil every so often to make sure that it has not dried out completely.
A bit of water perhaps once a month, just enough to keep the soil from cracking.

Here are some links to help you.

https://www.gardeningknowhow.com/ornamental/vines/honeysuckle/honeysuckle-vine-care.htm
https://www.gardeningknowhow.com/ornamental/vines/honeysuckle/prune-honeysuckle-plants.htm