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If they are not dug up, they can come back. The purple flower is just that, the flower of the plant. If you let it go, it will produce seeds.

This article will help you: https://www.gardeningknowhow.com/garden-how-to/seeds/planting-old-seeds.htm

Seeds for onions are normally only viable for a year.

Normally, you would replant them each year. These articles will help: https://www.gardeningknowhow.com/edible/vegetables/onion/growing-scallions.htm, https://www.gardeningknowhow.com/edible/vegetables/onion/harvesting-scallions.htm

You shouldn't allow them to bloom (or bolt) as this will lessen the plant's productivity and may cause a bitter taste. This article will help with bolting in members of the onion family: https://www.gardeningknowhow.com/edible/vegetables/onion/onion-bolting.htm For information on planting scallions, this article will help: https://www.gardeningknowhow.com/edible/vegetables/onion/growing-scallions.htm

If you are sure it is from one bulb, then I would not worry about thinning them. But if it is several bulbs bunched together, I would thin them so that the strongest plant will do the best.

Here are some articles on regrowing green onions that will help you:

In the garden:
http://www.veggiegardener.com/regrow-your-green-onions-for-maximum-use/

In water:
http://www.thekitchn.com/re-growing-green-onions-grow-your-scallions-back-on-your-windowsill-165274
http://www.17apart.com/2012/02/how-to-grow-green-onions-indefinitely.html