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You can keep them on the vine, but your vine will eventually began to dry up and die off.

The plants leaves may turn yellow from age, nutrient deficiency or infestation with pests or disease.
As the fruit ripens, the plant's older leaves at the base of the crown slowly die back. Although unsightly, this is a natural, harmless process. Cut off the dying leaves to help prevent the spread of rot and to improve air circulation around the plant.
As the fruit ripens, the plant's older leaves at the base of the crown slowly die back. Although unsightly, this is a natural, harmless process. Cut off the dying leaves to help prevent the spread of rot and to improve air circulation around the plant.
https://www.gardeningknowhow.com/edible/vegetables/squash/why-squash-leaves-turn-yellow.htm

Yes, thinning will allow larger pumkins and better quality.

They will not transplant well. This will either kill them, or prevent them from being able to produce fruit. I'm afraid they will have to stay there, and be pruned or maintained within their confines.

The plant will, naturally, send out runners, which will need to root in their own separate area. It would take multiple containers for each area that tries to send roots out. Now, it will continue to grow, but it will not be able to sustain a fruit in container, unfortunately.

Powdery mildew:
https://www.gardeningknowhow.com/edible/vegetables/pumpkin/powdery-mildew-on-pumpkins.htm