Old English docga, a late, rare word, used in at least one Middle English source in reference to a powerful breed of canine. The word forced out Old English hund (the general Germanic and Indo-European word, from root from PIE root *kwon-) by 16c. and subsequently was picked up in many continental languages (French dogue (16c.), Danish dogge, German Dogge (16c.)), but the origin remains one of the great mysteries of English etymology.
Many expressions -- a dog's life (c. 1600), go to the dogs (1610s), etc. -- reflect earlier hard use of the animals as hunting accessories, not pampered pets. In ancient times, "the dog" was the worst throw in dice (attested in Greek, Latin, and Sanskrit, where the word for "the lucky player" was literally "the dog-killer"), which plausibly explains the Greek word for "danger," kindynos, which appears to be "play the dog" (but Beekes is against this).
Slang meaning "ugly woman" is from 1930s; that of "sexually aggressive man" is from 1950s. Adjectival phrase dog-eat-dog attested by 1850s. Dog tag is from 1918. To dog-ear a book is from 1650s; dog-eared in extended sense of "worn, unkempt" is from 1894.
Notwithstanding, as a dog hath a day, so may I perchance have time to declare it in deeds. [Princess Elizabeth, 1550]
Phrase put on the dog "get dressed up" (1934) may look back to the stiff stand-up shirt collars that in the 1890s were the height of male fashion (and were known as dog-collars from at least 1883), with reference to collars worn by dogs. The common Spanish word for "dog," perro, also is a mystery word of unknown origin, perhaps from Iberian. A group of Slavic "dog" words (Old Church Slavonic pisu, Polish pies, Serbo-Croatian pas) likewise is of unknown origin.