lapdog n.

Old English læppa (plural læppan) "skirt or flap of a garment," from Proto-Germanic *lapp- (source also of Old Frisian lappa, Old Saxon lappo, Middle Dutch lappe, Dutch lap, Old High German lappa, German Lappen "rag, shred," Old Norse leppr "patch, rag"), of uncertain origin.

Sense of "lower front part of a shirt or skirt" led to that of "upper legs of seated person" (c. 1300). Used figuratively ("bosom, breast, place where someone or something is held and cherished") from late 14c., as in lap of luxury (which is first recorded 1802). To drop or dump something in someone's lap "shift a burden" is from 1962. From 15c.-17c. the word (often in plural) was a euphemism for "female pudendum," but this is not the source of lap dance, which is first recorded 1993.

To lap dance, you undress, sit your client down, order him to stay still and fully clothed, then hover over him, making a motion that you have perfected by watching Mister Softee ice cream dispensers. [Anthony Lane, review of "Showgirls," New Yorker, Oct. 16, 1995]

Lap-clap was old slang for "an act of coition" (c. 1600), in warning expressions to youth often paired with lip-clip "a kiss." Also compare slang Lapland "the society of women."

"quadruped of the genus Canis," Old English docga, a late, rare word, used in at least one Middle English source in reference specifically to a powerful breed of canine; other early Middle English uses tend to be depreciatory or abusive. Its origin remains one of the great mysteries of English etymology.

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.). 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. 

In reference to persons, by c. 1200 in abuse or contempt as "a mean, worthless fellow, currish, sneaking scoundrel." Playfully abusive sense of "rakish man," especially if young, "a sport, a gallant" is from 1610s. Slang meaning "ugly woman" is from 1930s; that of "sexually aggressive man" is from 1950s.  

Many expressions — a dog's life (c. 1600), go to the dogs (1610s), dog-cheap (1520s), etc. — reflect the earlier hard use of the animals as hunting accessories, not 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).

Notwithstanding, as a dog hath a day, so may I perchance have time to declare it in deeds. [Princess Elizabeth, 1550]

Meaning "something poor or mediocre, a failure" is by 1936 in U.S. slang. From late 14c. as the name for a heavy metal clamp of some kind. Dog's age "a long time" is by 1836. Adjectival phrase dog-eat-dog "ruthlessly competitive" is by 1850s. Phrase put on the dog "get dressed up" (1934) may be from comparison of dog collars 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).

And Caesar's spirit, ranging for revenge,

With Ate by his side come hot from Hell,

Shall in these confines with a monarch's voice

Cry Havoc! and let slip the dogs of war;

[Shakespeare, "Julius Caesar"]