Did any "exotic" breeds get shipped in from remote corners of the empire as especially extensive pets? Were there pet stores in the major cities? Did anyone make a living breeding and selling dogs? How would someone in the city even get a dog in the first place?
There are several examples of grave epitaphs for dogs:
"This is the tomb of the dog, Stephanos, who perished, Whom Rhodope shed tears for and buried like a human (vv. 4-5). I am the dog Stephanos, and Rhodope set up a tomb for me (v. 6)." Bryn Mawr Classical Review 2005.07.31
And others- "Thou who passest on this path, If haply thou dost mark this monument, Laugh not, I pray thee, though it is a dog's grave. Tears fell for me, and the dust was heaped above me By a master's hand." Select Epigrams from the Greek Anthology (1890) J.W. Mackail
Also the Naples Archaeology Museum has several life size dog statues in its collection from Pompeii, as well as a famous mosaic at a front door of a dog- I was just there a couple weeks ago, but I can't believe I don't have pictures of them!
Here is a photo of one of the dogs
I don't know of any primary sources for dog ownership/purchasing, but there may be some for sure. There are surviving representations of dogs wearing collars, so with the funerary monuments I think it is safe to make the leap that they were often well cared for.
I would imagine they were used as they commonly are today in rural areas: livestock herding/defense, hunting, and as warning systems against intruders.
In Pompeii there was a vogue for having paintings or mosaics of "guard dogs" at the entrance to your house saying "cave canem" (beware of dog). Here is a link with the most famous example along with a quote from Petronius about another example.
Edit: I decided to look up dogs in the Oxford Classical Dictionary and they have a pretty solid entry. The OCD, unlike a typical encyclopedia, uses primary material and gives a bibliography at the end of an entry. If you have any questions about Ancient Greece or Rome, this is the first place to look. Some libraries even have digital access.
dogs
were used by the Greeks and Romans as watchdogs; to guard livestock (but not to herd sheep or cattle); for hunting; and as pets. Odysseus, who, attacked by the dogs of Eumaeus, sits immobile until rescued by the animals' master, is pointed out as an example to the modern traveller in Greece; later in the Odyssey the king's own hound Argos (1d) provides one of the most moving moments in the poem, when he greets his disguised master and dies (Odyssey 14. 29–36, 17. 290–307). Among breeds mentioned by classical authors, the Laconian was particularly valued by Xenophon (1) for hunting; he distinguishes (Cynegeticus 3. 1–4. 9) between a larger Castorian and smaller vulpine, supposedly the result of crossing hounds with foxes. These may have approximated to the modern greyhound and whippet; for boar-hunting Indian, Cretan, and Locrian hounds were also necessary (Cynegeticus 10. 1). Indian hounds were probably mastiffs; better remembered today is the Molossian mastiff, first mentioned by Aristotle (HA 608a) and celebrated by Lucretius (5. 1062–71) who contrasts its behaviour when baying in an empty house, playing with its puppies, or flinching under the lash. Horace (Epodi 6. 5) undertakes to pursue his literary opponents ‘like a Molossian or tawny Laconian’. A small breed, like the Maltese terrier, appears on Attic vases; its ancient name is unknown. The Romans knew Italian breeds, like the Umbrian, described by Virgil (Aeneid 12. 749–55) in close pursuit of a stag, and from early imperial times were familiar with Celtic hounds, including those of Britain, the Agassaeans: small and ugly but strong, swift, and brave. Arrian (Cynegeticus 2–3) contrasts the slow and ugly Segusians, good only for tracking, with the swift and beautiful vertragi and adds a delightful account of his own favourite bitch Horme (‘Impulse’). Martial (14. 200) also praises the vertragus, as a soft-mouthed retriever. To Arrian (Cynegeticus 6. 1) no good hound is of a bad colour; in this he disagrees with Xenophon (Cynegeticus 4. 1–8) who holds that an unbroken colour indicates a savage strain and favours an admixture of tan. Columella (De re rustica 7. 12. 3) considers an unbroken colour advantageous for guard dogs; the shepherd will not mistake an all-white dog for a wolf in the twilight, while round the steading an all-black dog will seem formidable by day, and by night will be seen with difficulty. In fiction, the gigantic watchdog Scylax contributes notably to the confusion at Trimalchio's banquet (Petronius Satyrica 64, 72).
Bibliography
J. M. C. Toynbee, Animals in Roman Life and Art (1973).