Are dog breeds capitalized?

Typically, breeds are not capitalized in general-interest publications, like popular magazines and books and newspapers (see this from the NYT, for example). Common editorial practice follows the guidance of dictionaries like Merriam-Webster's and American Heritage. If you look up beagle or poodle in either of these sources, you will see that it is lower case. German shepherds, Portuguese water dogs, Newfoundland hounds, Dalmatians, and others that include a proper term do take capitals on those terms.

However, if you go to a site like the American Kennel Club's (www.akc.org), generally you'll see capitals used for all breeds, including Beagle and Poodle. This is a perfectly reasonable editorial decision, given that breeds and breeding are their entire focus.

In short, capitalization is an editorial choice.

Usually not, the one exception being "the Beagle" when referring to any given beagle, ie. the breed as a whole.