Myfanwy came through the Cardiff Space-Time Rift to Cardiff in the 21st century, where she was found hunting in the countryside and killing sheep at night. (TV: Everything Changes) Ianto Jones trapped Myfanwy in a warehouse, and managed to tranquilise her with Jack Harkness' help. (TV: Fragments)
Myfanwy was subsequently kept by Torchwood Three in the Hub as a mascot and pet, and became an essential part of their everyday lives; one time she even took part in a basketball game with the team. (TV: Cyberwoman)
When Lisa Hallett attacked the Torchwood team in the Hub, Jack sprayed Lisa with a barbecue sauce that caused Myfanwy to consider Lisa as food and to attack and injure her. (TV: Cyberwoman) Myfanwy survived her encounter with Lisa. (TV: Meat)
When Toshiko Sato unintentionally stopped the Hub's movement through time to avoid it being sent to the Sanctified by a Rift storm, Myfanwy was trapped out of sync with time with the Hub and Tosh. Myfanwy saved Tosh from falling to her death when the Harrowkind attacked, and carried her to the top of the tower. Myfanwy was subsequently returned with Tosh and the Hub. (COMIC: Rift War!)
When an Apatosaurus-like alien was accidentally brought into the Torchwood hub, Myfanwy tried to fight it, allowing Toshiko Sato to get to the armoury. After the alien was destroyed by a Jamolean lance, Myfanwy seemed to take pleasure in feeding on the burnt corpse. (PROSE: Pack Animals)
After investigating an alien spaceship that had crashed in a Welsh mine, Owen Harper and Ianto considered using Myfanwy for a lift back to Cardiff before deciding to use a bus instead. (AUDIO: The Last Beacon)
In an alternate timeline in which John Hart took control of the Torchwood Institute and became King of the British Empire, Torchwood Cardiff was likewise home to a Pterodactyl. John, however, was repeatedly offended by her bodily functions and had her shot just after introducing her to Gwen Cooper. (AUDIO: The Death of Captain Jack)
She was "house-trained", never attacking a member of staff, never making a mess or bringing in the local sheep she killed, and only ate food treated with a certain "barbecue sauce" which Suzie Costello had developed. (TV: Cyberwoman, PROSE: Pack Animals) Myfanwy also had a fondness for dark chocolate, (TV: Fragments) and would feed on birds that perched around the Hub. (PROSE: Another Life)
Behind the scenes
The name "Myfanwy" was initially a nickname created by the Torchwood cast, as shown in an episode [which?] of Torchwood Declassified. The name was later used in the pseudo-narrative Torchwood website, which featured an instant messaging conversation between Ianto and Owen about the pterodactyl's feeding habits. Until 2015 it was not used in-narrative. This changed with the release of Fall to Earth in which the name was finally used within the story.
The website also gave additional details about the creature. Myfanwy came and went from the Hub as she pleased, entering through a large opening in the ceiling; Jack Harkness trained her well enough so that she returned to the Hub upon hearing a signal from his wristwatch.