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Store your cured potatoes at about 40 degrees F. in a dark place. Too much light will turn them green. Never allow your potatoes to freeze.

Yes, you can do this. Just make sure a little of the stem pokes out the top of the straw as you fill in each time.

For the maximum crop, keep your potato vines well watered throughout the summer, but especially during the period when they are in flower, and immediately thereafter. Keep soil moist but not soggy. Water early in the day so that the foliage has time to dry completely before evening. When foliage turns yellow and dies back, discontinue watering.

If it is a blotchy red, it could be blight on the potatoes, but you would likely also see some unhealthiness in the foliage as well.

Another possibility is that the seed potatoes were mislabeled. The fungicide that was used to treat them may have masked the red color and had given them a lighter colored cast.

If you cut them back, you will be removing their ability to collect energy to store. Since they store excess energy in their tubers (aka potatoes) this means there would be fewer and smaller potatoes. It is best not to cut them back. Some people do mound up dirt or straw around the vines and this forces the plant to grow more tubers.

If you have not already done so, reading this article should help: https://www.gardeningknowhow.com/edible/vegetables/potato/how-and-when-to-harvest-potatoes.htm