I don't know that much about Euphorbias but I do know there are at least 1500 species within the genus so yours may be different from mine. The one I have (and love!) is wood spurge, Euphorbia amygdaloides (sp?). It is purple/green and has greenish/yellowish floral bracts. It does not rebloom but I do not deadhead it nor have I divided it because it reseeds like crazy which is great and not obnoxious at all. Mine seems to merrily enjoy it's abuse in dry shade and decent/rich woodland soil. I think a lot of them seed around if you let them.

As the previous post said, "euphorbia" is not an exact name. You have to know what species, variety, or hybrid you have. If you've got a native species, it's best not to deadhead, because birds and other critters will probably eat the seeds, plus the plant may selfseed. Also, the reason that many of us grow natives is that we want a more natural, less groomed look, and in nature, plants are not deadheaded.

With the exception of some weeds, there are very few species of euphorbia native to North America and no ornamental species I am aware of so I wouldn't worry too much about deadheading any natives to the detriment of wildlife. Any deadheading will not prompt rebloom on euphorbs anyway, however many of the evergreen species will not produce flowers (or cymes) on stalks that have already bloomed and removing them encourages the development of new growth for next season's flowers as well as reducing their propensity to reseed too freely. Herbaceous forms you can leave untouched until frost cuts them back, however many of these will reseed freely - in some cases rampantly - so removal of seed heads may be appropriate.