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If they were planted outside, they likely did not survive the winter. They are only hardy to zone 10 and will die elsewhere in the winter outside.

If they were brought inside for the winter, you can plant or put them back outdoors when the weather stays consistently above 50F. They will start to sprout when they are placed somewhere that is above 50F and they get regular water.

Caladiums will enter dormancy after a few months in leaf. I imagine they were in leaf for awhile at the nursery before you purchased them which is why they declined after a few weeks in your care. Caladiums, like other plants that grow from bulbs\corms\tubers, have a limited time to show off their splendor. The die-back process is normal and was not caused by anything you did - you did everything right.

For more information on caladium, please visit the following link:
https://www.gardeningknowhow.com/ornamental/bulbs/caladium/caladium-plant-care.htm

Yes, you can plant out your Caladiums, since in your growing zone the bulbs can stay in the ground year round.

https://www.gardeningknowhow.com/ornamental/bulbs/caladium/when-to-plant-caladium-bulbs.htm

Yes, it can. Please see this article:

https://www.gardeningknowhow.com/ornamental/bulbs/caladium/growing-caladiums-indoors.htm

They should have a small node or eye on the side which goes up, but this article will help
https://www.gardeningknowhow.com/ornamental/bulbs/caladium/caladium-plant-care.htm

It can vary considerably based on the depth they were planted, they type of soil, and how much light they receive. I have had mine vary over about a 5 week period from planting to seeing tips breaking through the soil. Be sure to not overwater, especially in the area where they haven't come up yet as their water needs are greater once they start developing their foliage. They are a wonderful plant to add foliage colour.