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Your best option will be to cure and store these inside, if the area is too wet to do so outside. This article will help: https://www.gardeningknowhow.com/edible/vegetables/sweet-potato/sweet-potato-storage-tips.htm

As long as you have a horticultural light fixture, then you can grow this indoors. Mine currently reside under Powerful LED lighting along with the rest of my garden. In fact, I just harvested mine, and started a new plant from one of the tips of the last. You can even do this with the ornamental varieties. The leaves are also edible, as long as you cook them first.

If they are in a peat mixture, then it is likely that they do not have any nutrients at all. Any well balanced fertilizer will be fine. If you were to replace the peat with potting soil, then you will not need to fertilize at all. They are not heavy feeders, as they have edible root storage for nutrients. (These are sweet potatoes, essentially!) If you choose to leave them in the peat, then they will need a weekly half strength liquid fertilizer. Any 10-10-10 will be fine

You can! You will be best doing this in container, though It would help to start them indoors well before the soil warms up under horticultural lighting. This will allow you to plant out a fully mature plant. This means that you will get a good harvest, even though your warm season isn't long enough for sweet potatoes.

This article will help you to grow sweet potatoes in container: https://www.gardeningknowhow.com/edible/vegetables/sweet-potato/sweet-potato-container-crops.htm

Cabbage Loopers may be the culprit.

This article will help you.
https://www.gardeningknowhow.com/edible/vegetables/cabbage/cabbage-looper-control.htm

They should be ready to harvest in about 150 days. Here are tips for growing in containers:

https://www.gardeningknowhow.com/edible/vegetables/sweet-potato/sweet-potato-container-crops.htm