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Four o'clocks are often mid-summer bloomers. since I don't know where you are, it may be that it is not quite time for blooms there. You mention the foliage looks good, if this is the result of lots of nitrogen fertilizer, the plant may be directing energy into foliage instead of flowers. If so, stop fertilizing, keep them watered and blooms may soon appear.

Yes, it helps to deadhead these plants, especially if you don't want them popping up everywhere else in the garden, as they are self-seeding in many places.

You do not need to dig up these roots if you are concerned with winter damage. They should naturally die back over winter but will regrow once spring returns. In fact, some of the dropped seeds may even sprout new plants in various areas of the garden.

As for root division, you should wait for spring to divide these plants and that's if they are outgrowing their boundaries and becoming crowded. You would simply dig up the clump and use a spade shovel (or whatever) to split the clump into as many sections as you require, each having their own root system. Then replant elsewhere.

The plant is possibly "Four O'Clocks" or Marabilis Jalapa. This article should help: http://www.gardensablaze.com/Perennials/Perennials4oclock.htm

This blog post has a picture of the seedlings:
http://xploreandxpress.blogspot.com/2012/01/organic-gardening-four-o-clock-flowers.html

Absolutely, thin them. I'm not sure how well they will transplant in hot weather, but you could collect some mature seed and let them germinate where you want them.